Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps

Abstract

Let $\Gamma$ be a geometrically finite discrete subgroup in ${\text {SO}}(d+1,1)^{\circ }$ with parabolic elements. We establish exponential mixing of the geodesic flow on the unit tangent bundle ${\text {T}}^1(\Gamma \backslash {\mathbb {H}}^{d+1})$ with respect to the Bowen–Margulis–Sullivan measure, which is the unique probability measure on ${\text {T}}^1(\Gamma \backslash {\mathbb {H}}^{d+1})$ with maximal entropy. As an application, we obtain a resonance-free region for the resolvent of the Laplacian on $\Gamma \backslash {\mathbb {H}}^{d+1}$. Our approach is to construct a coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator.

1 Introduction

1.1 Exponential mixing of the geodesic flow

Let ${\mathbb {H}}^{d+1}$ be the hyperbolic $(d+1)$-space. Let $G={\text {SO}}(d+1,1)^{\circ }$, which is the group of orientation preserving isometries of ${\mathbb {H}}^{d+1}$. Let $\Gamma <G$ be a non-elementary, torsion-free, geometrically finite discrete subgroup with parabolic elements. Denote by $\delta$ the critical exponent of $\Gamma$, which is defined as the abscissa of convergence of the Poincaré series $\sum _{\gamma \in \Gamma }e^{-sd(o,\gamma o)}$. Set $M=\Gamma \backslash {\mathbb {H}}^{d+1}$, so M contains cusps. We consider the geodesic flow $({\mathcal {G}}_t)_{t\in {\mathbb {R}}}$ acting on the unit tangent bundle ${\text {T}}^1(M)$ over M. The invariant measure for the flow we will work with is the Bowen–Margulis–Sullivan measure $m^{{\text {BMS}}}$, which is supported on the non-wandering set of the geodesic flow and is known to be the unique probability measure with maximal entropy $\delta$ [37].

Our main result is establishing exponential mixing of the geodesic flow.

Theorem 1.1

The geodesic flow is exponentially mixing with respect to $m^{{\text {BMS}}}$: there exists $\eta >0$ such that for any functions $\phi , \psi \in C^1({\text {T}}^1(M))$ and any $t>0$, we have

$$\begin{aligned} \int _{{\text {T}}^1(M)} \phi \cdot \psi \circ {{\mathcal {G}}}_t\ \mathrm {d}m^{{\text {BMS}}}=m^{{\text {BMS}}}(\phi ) m^{{\text {BMS}}}(\psi )+O(\Vert \phi \Vert _{C^1} \Vert \psi \Vert _{C^1}e^{-\eta t}), \end{aligned}$$

where $\Vert \cdot \Vert _{C^1}$ is the $C^1$-norm with respect to the Riemannian metric on ${\text {T}}^1(M)$.

For a geometrically finite discrete subgroup $\Gamma$, Sullivan [49] proved the ergodicity of the geodesic flow with respect to $m^{{\text {BMS}}}$ and Rudolph [43] proved that the geodesic flow is mixing with respect to $m^{{\text {BMS}}}$. When $\delta >d/2$, Theorem 1.1 was proved by Mohammadi–Oh [33] and Edwards–Oh [19] using the representation theory of $L^2(M)$ and the spectral gap of Laplace operator [28]. When $\Gamma$ is convex cocompact, i.e., geometrically finite without parabolic elements, Theorem 1.1 and its corollaries were proved by Naud [34], Stoyanov [47] and Sarkar–Winter [44] building on the work of Dolgopyat [18]. Therefore, the main contribution of our work lies in the groups with small critical exponent and with parabolic elements, completing the story of exponential mixing of the geodesic flow on a geometrically finite hyperbolic manifold.

Using Roblin’s transverse intersection argument [35, 36, 42], we obtain the decay of matrix coefficients (Theorem 9.1) from Theorem 1.1. Theorem 1.1 and 9.1 are known to have many immediate applications in number theory and geometry. To name a few, see [32] for counting closed geodesics, [25] for shrinking target problems and [7] for some general counting results.

1.2 Resonance-free region

Recall $M=\Gamma \backslash {\mathbb {H}}^{d+1}$. Consider the Laplace operator $\Delta _M$ on M. Lax and Phillips completely described its spectrum on $L^2(M)$ when M has infinite volume [28]. The half line $[d^2/4,\infty )$ is the continuous spectrum and it contains no embedded eigenvalues. The rest of the spectrum (point spectrum) is finite and starting at $\delta (d/2-\delta )$ if $\delta >d/2$ and is empty if $\delta \le d/2$. Let S be the set of eigenvalues of $\Delta _M$. The resolvent of the Laplacian

$$\begin{aligned} R_M(s)=(\Delta _M-s(d-s))^{-1}:L^2(M)\rightarrow L^2(M) \end{aligned}$$

is well-defined and analytic on$\{\Re s>d/2,\ s(d-s)\notin S\}$. Guillarmou and Mazzeo showed that $R_M(s)$ has a meromorphic continuation to the whole complex plane as an operator from $C^\infty _c(M)$ to $C^\infty (M)$ with poles of finite rank [21]. These poles are called resonances. Patterson showed that on the line $\Re s= \delta$, the point $s=\delta$ is the unique pole of $\Gamma (s-\frac{d}{2}+1)R_M(s)$ and it is a simple pole [39]. We use Theorem 1.1 to further obtain a resonance-free region.

Theorem 1.2

There exists $\eta >0$ such that on the half-plane $\Re s>\delta -\eta$, $s=\delta$ is the only resonance for the resolvent $R_M(s)$ if $\delta \notin d/2-{\mathbb {N}}_{\ge 1}$; otherwise, $R_M(s)$ is analytic on $\Re s>\delta -\eta$.

In the convex cocompact case, a resonance-free region of the resolvent is closely related to a zero free region of the Selberg zeta function. But in the geometrically finite case, such relation is not well understood except for the surface case.

1.3 On the proof of the main theorem

The proof of Theorem 1.1 can be reduced to the case when $\Gamma$ is Zariski dense and then the proof falls into two parts: we code the geodesic flow and prove a Dolgopyat-type spectral estimate for the corresponding transfer operator. Ultimately, the obstructions to applying Dolgopyat’s original argument in our context are purely technical, but to overcome these obstructions in any context is the heart of the matter.

To prove exponential mixing using the symbolic-dynamic approach of Dolgopyat, one approach is to construct a section to the flow. In sum, one seeks a 2d submanifold S in ${\text {T}}^1(M)$ transversal to the geodesic flow which is a Poincaré section, on which the return map can be tightly organized. The challenge lies in that it is required to find a return map R defined on a full measure subset $S_0$ of S, such that the map $F(v):={{\mathcal {G}}}_{R(v)}(v)$, $v\in S_0$, on $S_0$ is hyperbolic and can be modelled on a full shift of countable many symbols.

We overcome this difficulty by connecting the return map on $S_0$ to an expanding map on the boundary $\partial {\mathbb {H}}^{d+1}$. The precise description of the expanding map on the boundary is as follows. We consider the upper-half space model for ${\mathbb {H}}^{d+1}$ and without loss of generality, we may assume that $\infty$ is a parabolic fixed point of $\Gamma$. Let ${\text {Stab}}_{\infty }(\Gamma )$ be the group of stabilizers of $\infty$ in $\Gamma$ and $\Gamma _{\infty }$ be a maximal normal abelian subgroup in ${\text {Stab}}_{\infty }(\Gamma )$. Set $\Delta _{0}:=\Delta _{\infty }$ to be a fundamental domain of $\Gamma _{\infty }$ in $\partial {\mathbb {H}}^{d+1}\backslash \{\infty \}$ (see Sect. 2.3 for details). Denote by $\Lambda _{\Gamma }$ the limit set of $\Gamma$ and $\mu$ the Patterson–Sullivan measure, which is a finite measure supported on $\Lambda _{\Gamma }$.

Proposition

4.1 There are constants $C_1>0$, $\lambda ,\,\epsilon _0\in (0,1)$, a countable collection of disjoint, open subsets $\{\Delta _j\}_{j\in {\mathbb {N}}}$ in $\Delta _0$ and an expanding map $T:\sqcup _j\Delta _j\rightarrow \Delta _0$ such that:

• 1. $\sum _{j}\mu (\Delta _j)=\mu (\Delta _0)$.

• 2. For each j, there is an element $\gamma _j\in \Gamma$ such that $\Delta _j=\gamma _j \Delta _0$ and $T|_{\Delta _j}=\gamma _j^{-1}$.

• 3. Each $\gamma _j$ is a uniform contraction: $|\gamma _j'(x)|\le \lambda$ for all $x\in \Delta _0$.

• 4. For each $\gamma _j$, $|\mathrm D(\log |\gamma _j'|)(x)|<C_1$ for all $x\in \Delta _0$, where $\mathrm D(\log |\gamma _j'|)(x)$ is the differential of the map $z\mapsto \log |\gamma _j'(z)|$ at x.

• 5. Let R be the function on $\sqcup _j \Delta _j$ given by $R(x)=\log |\mathrm DT(x)|$. Then $\int e^{\epsilon _o R}\mathrm {d}\mu <\infty .$

The last property is known as the exponential tail property. Moreover, we show that the coding satisfies the uniform nonintegrable condition (UNI) (Lemma 4.5). We use Proposition 4.15 as a bridge to connect the geodesic flow $({\mathcal {G}}_t)_{t\in {\mathbb {R}}}$ on ${\text {T}}^1(M)$ and the expanding map T on $\Delta _0$. We show that the geodesic flow is a factor of a hyperbolic skew product flow constructed using T.

The construction of the coding starts with the following observation: locally, in a neighborhood of a parabolic fixed point, we can use the structure of the parabolic fixed points to find a “flower” centered at this parabolic fixed point, and the “flower” can be partition into a countable union of open sets of the form $\gamma (\Delta _0)$ for some $\gamma \in \Gamma$. Once we have the algorithm to find the partition for local regions, we still face the question of how to patch these flowers together. We introduce an inductive algorithm to find pairwise disjoint flowers.

But the bulk of the work lies in proving the exponential tail property. We show that this follows from Proposition 6.15 which says that the measure of the set that has not been partitioned at time n decays exponentially. At the time n, the remaining part is a sheet with many holes, consisting of “flowers”; while the Patterson–Sullivan measure is a measure supported on the fractal limit set, and we have limited knowledge of the regularity of this measure. It is interesting to figure out how to use minimal tools to get the required estimate.

When the non-wandering set of the geodesic flow is compact, the coding is well-studied and we have, for example, the Bowen–Series’ coding [6], Bowen’s coding [9] and Ratner’s coding [41]. When manifolds contain cusps, only some partial knowledge is available. Dal’bo–Peigné [15, 16] and Babillot–Peigné [3] provided the coding for generalized Schottky groups. Stadlbauer [45] and Ledrappier-Sarig [31] provided the coding for non-uniform lattices in ${\text {SO}}(2,1)^{\circ }$. They made use of the fact that such a discrete subgroup is a free group and has a nice fundamental domain in ${\mathbb {H}}^2$. Our coding works for general geometrically finite discrete subgroups with parabolic elements and is partly inspired by the works of Lai–Sang Young [51] and Burns–Masur–Matheus–Wilkinson [11].

In a forthcoming joint work with Sarkar [29], we establish the exponential mixing for frame flows on geometrically finite hyperbolic manifolds with cusps. We prove this by using the coding of the geodesic flow constructed in this paper and then performing a frame flow version of Dolgopyat’s method. The crucial cancellations of the summands of the transfer operators twisted by holonomy are obtained from the local non-integrability condition and the non-concentration property of Sarkar–Winter [44]. But the challenge in the presence of cusps is that the latter holds only on a certain good subset. This is resolved by a large deviation property for symbolic recurrence to the good subset, which is inspired by the work of Tsujii–Zhang [50]. It is proved by studying the combinatorics of cusp excursions and showing an effective renewal theorem, as in the work of Li [30], which uses the spectral gap of the transfer operator for the geodesic flow in Proposition 7.3.

In [22], using the coding of Schottky groups, Guillopé–Lin–Zworski were able to study the Selberg zeta function through a dynamical zeta function. They gave a simple proof of the analytic continuation and a growth estimate of the Selberg zeta function. Hopefully, the coding constructed in our work will be helpful in the study of the Selberg zeta function for higher dimensional geometrically finite manifold.

Other applications include the Fourier decay of the Patterson–Sullivan measure. In [4], Bourgain–Dyatlov proved Fourier decay of Patterson-Sullivan measures for convex cocompact Fuchsian groups. The first step of their proof is to use the coding of the limit set to construct an appropriate transfer operator. With our coding available, it is very likely to generalize the Fourier decay to geometrically finite discrete subgroups with parabolic elements.

Our proof of obtaining a Dolgopyat-type spectral estimate is influenced by the one in [1, 2, 5, 18, 34, 47]. The key of Dolgopyat’s approach is to estimate the decay of certain oscillatory integrals against the fractal Patterson-Sullivan measure: for function f of the form $\sum _{j\in J}\exp (ib\tau _j(x))$, where b is a real number, $\tau _j\in C^2(\Delta _{0})$ and J is some index set, we have $|\int f\,d\mu |$ is bounded by some negative power of |b|. We successfully attain this estimate by combining dynamics and the regularity properties of the Patterson–Sullivan measure, which we think are the essential ingredients to gain the decay.

Another possible argument is to analyze each $\int \exp (ib\tau _j)\ d\mu$ and show the decay. Such a result is known as the Fourier decay of the Patterson–Sullivan measure. This is especially challenging when the critical exponent $\delta$ is small. In [23], Jordan–Sahlsten proved the Fourier decay of some fractal measures. Their idea is to approximate the fractal measure by the Lebesgue measure and use the Fourier decay of the Lebesgue measure, which is well-studied. But this approximation is sensitive to the Hausdorff dimension of the fractal sets. A similar idea also appears in a preprint by Kahlil [26]. It is unclear whether their approach provides an alternative way to establish the Fourier decay of fractal measures without dimension restriction.

There are works trying to use anisotropic Banach spaces to prove exponential mixing. The key is to show there exists $\epsilon >0$ such that the strip $\{-\epsilon<\Re s<0 \}$ is free of Pollicott–Ruelle resonances. For a geometrically finite discrete subgroup with the critical exponent $\delta <d/2$, it might happen that there are resonances with large imaginary parts and real parts close to zero. Recently, there is a work in progress of Gouëzel–Tapie–Schapira on the Pollicott–Ruelle resonances for SPR manifolds, which include geometrically finite manifolds. They show the resonances are discrete, but it is not clear whether one can use this property to attain the required resonance-free region.

Organization of the paper.

• In Sect. 2, we gather the basic facts and preliminaries about hyperbolic spaces, geometrically finite discrete subgroups, the structure of cusps, Patterson-Sullivan measure, and Bowen–Margulis–Sullivan measure.

• In Sect. 3, we prove that Theorem 1.1 can be reduced to Zariski dense case.

• In Sect. 4, we state the results of the coding (Proposition 4.1, Lemma 4.5, 4.8). We construct a hyperbolic skew product flow and state the result that it is exponential mixing (Theorem 4.13). We show that the geodesic flow on ${\text {T}}^1(M)$ is a factor of this hyperbolic skew product flow (Proposition 4.15) and deduce the exponential mixing of the geodesic flow from Theorem 4.13.

• In Sect. 5, we provide an explicit description of the action of an element $\gamma \in \Gamma$ on $\partial {\mathbb {H}}^{d+1}$ and the estimate on the norm of the derivative of $\gamma$ (Sect. 5.1). We list the basics for the multi-cusp case (Sect. 5.2). The doubling property and the friendliness of Patterson-Sullivan measure are proved in Sects. 5.3 and 5.4.

• In Sect. 6, we start with the construction of the coding for one cusp case, which is also the first step for multi-cusp case. The main result is exponential decay of the remaining set (Proposition 6.15). Section 6.3–6.6 are devoted to the proof the Proposition 6.15. The coding for the multi-cusp case will be provided in Sects. 6.7 and 6.8. The results of the coding (Proposition 4.1, Lemma 4.5, 4.8) will be proved in Sect. 6.7-6.9.

• In Sect. 7, we prove a Dolgopyat-type spectral estimate for the corresponding transfer operator and the main result is an $L^2$-contraction proposition (Proposition 7.3).

• In Sect. 8, we finish the proof of Theorem 4.13.

• In Sect. 9, we prove the application of obtaining a resonance-free region for the resolvent $R_{M}(s)$ (Theorem 1.2).

Notation. In the paper, given two real functions f and g, we write $f\ll g$ if there exists a constant $C>0$ only depending on $\Gamma$ such that $f\le Cg$. We write $f\approx g$ if $f\ll g$ and $g\ll f$.

2 Preliminary of hyperbolic spaces and PS measure

2.1 Hyperbolic spaces

We will use the upper-half space model for ${\mathbb {H}}^{d+1}$:

$$\begin{aligned} {\mathbb {H}}^{d+1}=\{x=(x_1,\ldots ,x_{d+1})\in {\mathbb {R}}^{d+1}:\,x_{d+1}>0 \}. \end{aligned}$$

Let $o=(0,\ldots ,0,1)\in {\mathbb {H}}^{d+1}$. For $x\in {\mathbb {H}}^{d+1}$, write h(x) for the height of the point x, which is the last coordinate of x. The Riemannian metric on ${\mathbb {H}}^{d+1}$ is given by

$$\begin{aligned} \mathrm {d}s^2=\frac{\mathrm {d}x_1^2+\cdots +\mathrm {d}x_{d+1}^2}{x_{d+1}^2}. \end{aligned}$$

Let $\partial {\mathbb {H}}^{d+1}$ be the visual boundary. On $\partial {\mathbb {H}}^{d+1}={\mathbb {R}}^d\cup \{\infty \}$, we have the spherical metric, denoted by $d_{{\mathbb {S}}^d}(\cdot ,\cdot )$. We also have the Euclidean metric, denoted by $d_{E}(x,x')$ or $|x-x'|$ for any $x,x'\in \partial {\mathbb {H}}^{d+1}$. This metric will be used most frequently; we will simply write $d(\cdot , \cdot )$ when there is no confusion.

For $g\in G$, it acts on $\partial {\mathbb {H}}^{d+1}$ conformally. For $x\in \partial {\mathbb {H}}^{d+1}$, let $|g'(x)|$ be the linear distortion of the conformal action of g at x with respect to the Euclidean metric. It is also the norm of the derivative seen as a linear map on tangent spaces. Let $|g'(x)|_{{\mathbb {S}}^d}$ be the norm with respect to the spherical metric. We have the relation

$$\begin{aligned} |g'(x)|_{{\mathbb {S}}^d}=\frac{1+|x|^2}{1+|gx|^2}|g'(x)|. \end{aligned}$$

(2.1)

Another formula for $|g'(x)|_{{\mathbb {S}}^d}$ is

$$\begin{aligned} |g'(x)|_{{\mathbb {S}}^d}=e^{-\beta _x(g^{-1}o,o)}, \end{aligned}$$

where $\beta _x(\cdot ,\cdot )$ is the Busemann function given by $\beta _x(z,z')=\lim _{t\rightarrow +\infty }d(z,x_t)-d(z',x_t)$ with $x_t$ an arbitrary geodesic ray tending to x.

We denote ${\mathbb {H}}^{d+1}\cup \partial {\mathbb {H}}^{d+1}$ by $\overline{{\mathbb {H}}^{d+1}}$.

2.2 Geometrically finite discrete subgroups

Let $\Gamma$ be a torsion-free, non-elementary discrete subgroup in G. We list some basics of geometrically finite discrete subgroups.

The limit set of $\Gamma$ is the set $\Lambda _{\Gamma }$ of all the accumulation points of an orbit $\Gamma x$ for some $x\in {\mathbb {H}}^{d+1}$. As we assume $\Gamma$ is torsion-free, $\Lambda _{\Gamma }$ is contained in $\partial {\mathbb {H}}^{d+1}$. The convex hull, $\text {hull}(\Lambda _{\Gamma })$, of $\Lambda _{\Gamma }$ is the smallest convex subset in ${\mathbb {H}}^{d+1}$ which contains all the geodesics connecting any two distinct points of $\Lambda _{\Gamma }$. The convex core of M is $C(M)=\Gamma \backslash \text {hull}(\Lambda _{\Gamma })\subset M$.

A limit point $x\in \Lambda _{\Gamma }$ is called conical if there exists a geodesic ray tending to x and a sequence of elements $\gamma _n\in \Gamma$ such that $\gamma _no$ converges to x, and the distance between $\gamma _no$ and the geodesic ray is bounded. A subgroup $\Gamma '$ of $\Gamma$ is called parabolic if $\Gamma '$ fixes only one point in $\partial {\mathbb {H}}^{d+1}$. A point $x\in \Lambda _\Gamma$ is called a parabolic fixed point if its stabilizer in $\Gamma$, $\mathrm{{Stab}}_{\Gamma }(x)$, is parabolic. A parabolic fixed point is called bounded parabolic if the quotient $\mathrm {Stab}_{\Gamma }(x)\backslash (\Lambda _{\Gamma }-\{x\})$ is compact.

A horoball based at $x\in \partial {\mathbb {H}}^{d+1}$ is the set $\{y\in {\mathbb {H}}^{d+1}:\,\ \beta _x(y,o)<t \}$ for some $t\in {\mathbb {R}}$. The boundary of a horoball is called a horosphere. We call a horoball H based at a parabolic fixed point $x\in \Lambda _{\Gamma }$ a horocusp region, if we have $\gamma H\cap H=\emptyset$ for any $\gamma \in \Gamma -\mathrm {Stab}_{\Gamma }(x)$. Then the image of H in M under the quotient map, $\Gamma \backslash \Gamma H$, is isometric to $\mathrm {Stab}_{\Gamma }(x)\backslash H$ and is called a proper horocusp of M.

Definition 2.2

(Geometrically finite discrete subgroup [8, 40]) A non-elementary discrete subgroup $\Gamma <{\text {SO}}(d+1,1)^{\circ }$ is called geometrically finite if it satisfies one of the following equivalent conditions:

• (i) There is a (possibly empty) finite union V of proper horocusps of M, with disjoint closures, such that $C(M)-V$ is compact.

• (ii) Every limit point of $\Gamma$ is either conical or bounded parabolic.

2.3 Structure of cusps

Assume that $\Gamma$ is a geometrically finite discrete subgroup with parabolic elements and $\infty$ is a parabolic fixed point of $\Gamma$. Let $\Gamma _\infty ^{'}=\mathrm {Stab}_\Gamma (\infty )$ be the parabolic subgroup of $\Gamma$ fixing $\infty$. Then $\Gamma _\infty ^{'}$ acts on ${\mathbb {R}}^d$, part of $\partial {\mathbb {H}}^{d+1}$, isometrically with respect to the Euclidean metric. The following is a result of Bieberbach (see [21, Page 5] or [8, Section 2.2]).

Lemma 2.3

(Bieberbach) Consider the action of $\Gamma _{\infty }'$ on ${\mathbb {R}}^d$. Then there exist a maximal normal abelian subgroup $\Gamma _\infty \subset \Gamma _\infty ^{'}$ of finite index and an affine subspace $Z\subset {\mathbb {R}}^d$ of dimension k, invariant under $\Gamma _{\infty }'$, such that $\Gamma _{\infty }$ acts as a group of translations of rank k on Z. If ${\mathbb {R}}^d=Y\times Z$ is an orthogonal decomposition, with $Y\simeq {\mathbb {R}}^{d-k}$ and associated coordinates (y, z), then we can write each element $\gamma \in \Gamma _\infty ^{'}$ in the form

$$\begin{aligned} \gamma (y,z)=(A_\gamma y, R_\gamma z+b_\gamma ), \,\,\, b_{\gamma }\in {\mathbb {R}}^{k},\,\,\,A_\gamma \in O(n-k),\ R_\gamma \in O(k) \end{aligned}$$

where for each $\gamma$, $R_{\gamma }^m={\text {Id}}$ for some $m\in {\mathbb {N}}$, with $m=1$ if $\gamma \in \Gamma _\infty$.

The dimension k is called the rank of the parabolic fixed point $\infty$.

Fix an orthogonal decomposition ${\mathbb {R}}^d=Y\times Z\simeq {\mathbb {R}}^{d-k}\times {\mathbb {R}}^k$. As $\Gamma _{\infty }$ acts on ${\mathbb {R}}^k$ as a group of translations, it admits a fundamental region $\Delta _{\infty }'$ which is an open k-dimensional parallelotope in ${\mathbb {R}}^k$. Since $\Gamma$ is geometrically finite, $\infty$ is a bounded parabolic fixed point. By definition, the quotient $\Gamma _{\infty }'\backslash (\Lambda _{\Gamma }-\{\infty \} )$ is compact; the quotient $\Gamma _{\infty }\backslash (\Lambda _{\Gamma }-\{\infty \} )$ is also compact as $\Gamma _{\infty }$ is a finite index subgroup of $\Gamma _{\infty }'$. Therefore, there exists a constant $C>0$ such that the set $B_Y(C)=\{y\in {\mathbb {R}}^{d-k}:\, |y|< C \}$ in ${\mathbb {R}}^{d-k}$ has the property that Fig. 1

$$\begin{aligned} \Lambda _{\Gamma }\subset \{\infty \}\cup \left( \cup _{\gamma \in \Gamma _\infty }\gamma \left( \overline{B_Y(C/2)\times \Delta '_{\infty }}\right) \right) . \end{aligned}$$

Definition 2.4

We call the open set $\Delta _{\infty }:=B_Y(C)\times \Delta _\infty '$ a fundamental region for the parabolic fixed point $\infty$.

2.4 PS measure and BMS measure

Patterson–Sullivan measure. Recall $\delta$ is the critical exponent of $\Gamma$. Patterson [38] and Sullivan [48] constructed a $\Gamma$-invariant conformal density $\{\mu _y \}_{y\in {\mathbb {H}}^{d+1}}$ of dimension $\delta$ on $\Lambda _{\Gamma }$, which is a set of finite Borel measures such that for any $y,z\in {\mathbb {H}}^{d+1}$, $x\in \partial {\mathbb {H}}^{d+1}$ and $\gamma \in \Gamma$,

$$\begin{aligned} \frac{\mathrm {d}\mu _y}{\mathrm {d}\mu _z}(x)=e^{-\delta \beta _{x}(y,z)}\,\,\,\text {and}\,\,\, (\gamma )_{*}\mu _y=\mu _{\gamma y}, \end{aligned}$$

(2.5)

where $\gamma _{*}\mu _{y}(E)=\mu _{y}(\gamma ^{-1}E)$ for any Borel subset E of $\partial {\mathbb {H}}^{d+1}$. This family of measures is unique up to homothety, and the action of $\Gamma$ on $\partial {\mathbb {H}}^{d+1}$ is ergodic relative to the measure class defined by these measures (Fig. 1).

As $\mu _y$’s are absolutely continuous with respect to each other, for most of the paper, we will consider $\mu _{o}$ and denote it by $\mu$ for short. We call it the Patterson–Sullivan measure (or PS measure). The following quasi-invariance property of the PS measure will be frequently used: for any Borel subset E of $\partial {\mathbb {H}}^{d+1}$ and any $\gamma \in \Gamma$,

$$\begin{aligned} \mu (\gamma E)=\int _E|\gamma '(x)|_{{\mathbb {S}}^d}^\delta \mathrm {d}\mu (x). \end{aligned}$$

(2.6)

Bowen–Margulis–Sullivan measure. Let $\partial ^2 ({\mathbb {H}}^{d+1})=\partial {\mathbb {H}}^{d+1}\times \partial {\mathbb {H}}^{d+1}-\text {Diagonal}$. The Hopf parametrization of ${\text {T}}^1({\mathbb {H}}^{d+1})$ as $\partial ^2 ({\mathbb {H}}^{d+1})\times {\mathbb {R}}$ is given by

$$\begin{aligned} v\mapsto (x,x_-,s=\beta _{x}(o,v_*)), \end{aligned}$$

where x (resp. $x_-$) is the forward endpoint (resp. backward) endpoint of v under the geodesic flow, and $v_*\in {\mathbb {H}}^{d+1}$ is the based point of v. The geodesic flow on ${\text {T}}^1({\mathbb {H}}^{d+1})$ is represented by the translation on ${\mathbb {R}}$-coordinate.

Fig. 1

Here $\infty$ is a parabolic fixed point of rank 2 in $\partial {\mathbb {H}}^4$. The intersection $\Lambda _{\Gamma }\cap {\mathbb {R}}^3$ has bounded distance to ${\mathbb {R}}^2$

The Bowen–Margulis–Sullivan measure (or BMS measure) on ${\text {T}}^1({\mathbb {H}}^{d+1})$ is defined by

$$\begin{aligned} \mathrm {d}{\tilde{m}}^{{\text {BMS}}}(x,x_-,s)=e^{\delta \beta _{x}(o,x_*)} e^{\delta \beta _{x_-}(o,x_*)}\mathrm {d}\mu (x) \mathrm {d}\mu (x_-)\mathrm {d}s, \end{aligned}$$

where $x_*$ is the based point of the unit tangent vector given by $(x,x_-,s)$. It is invariant under the geodesic flow ${{\mathcal {G}}}_t$ from the definition. The group $\Gamma$ acts on $\partial ^2 ({\mathbb {H}}^{d+1})\times {\mathbb {R}}$ by

$$\begin{aligned} \gamma (x,x_-,s)=(\gamma x, \gamma x_-,s-\beta _{x}(o,\gamma ^{-1}o)). \end{aligned}$$

This formula, together with (2.5), implies that ${\tilde{m}}^{{\text {BMS}}}$ is left $\Gamma$-invariant; hence ${\tilde{m}}^{{\text {BMS}}}$ induces a measure $m^{{\text {BMS}}}$ on ${\text {T}}^1(M)$, which is the Bowen–Margulis–Sullivan measure on ${\text {T}}^1(M)$. For geometrically finite discrete subgroups, Sullivan showed that $m^{{\text {BMS}}}$ is finite and ergodic with respect to the action of the geodesic flow [49]. Otal and Peigné showed that $m^{{\text {BMS}}}$ is the unique measure supported on the non-wandering set of the geodesic flow with maximal entropy $\delta$ [37]. After normalization, we suppose that $m^{{\text {BMS}}}$ is a probability measure.

3 Reduction to Zariski dense case

The group ${\text {SO}}(d+1,1)$ is Zariski closed and connected and the subgroup ${\text {SO}}(d+1,1)^\circ$ is its analytic connected component containing identity. For a subgroup $\Gamma$ of ${\text {SO}}(d+1,1)^\circ$, it is said to be Zariski dense in ${\text {SO}}(d+1,1)^\circ$ if it is Zariski dense in ${\text {SO}}(d+1,1)$. The proof of Theorem 1.1 can be reduced to Zariski dense case.

Theorem 3.1

Assume that $\Gamma <{\text {SO}}(d+1,1)^{\circ }$ is a Zariski dense, torsion-free, geometrically finite subgroup with parabolic elements. The geodesic flow $({\mathcal {G}}_t)_{t\in {\mathbb {R}}}$ on ${\text {T}}^1(M)$ is exponentially mixing with respect to $m^{{\text {BMS}}}$: there exists $\eta >0$ such that for any functions $\phi , \psi \in C^1({\text {T}}^1(M))$ and any $t>0$, we have

$$\begin{aligned} \int _{{\text {T}}^1(M)} \phi \cdot \psi \circ {{\mathcal {G}}}_t\ \mathrm {d}m^{{\text {BMS}}}=m^{{\text {BMS}}}(\phi ) m^{{\text {BMS}}}(\psi )+O(\Vert \phi \Vert _{C^1} \Vert \psi \Vert _{C^1}e^{-\eta t}). \end{aligned}$$

From Theorem 3.1to Theorem1.1 Suppose $\Gamma$ is not Zariski dense. Let H be the Zariski closure of $\Gamma$ in ${\text {SO}}(d+1,1)$ and let $H_1$ be the Zariski connected component of H containing the identity. Let $\Gamma _1=\Gamma \cap H_1$. Then $\Gamma _1$ is a finite index subgroup of $\Gamma$ and the Zariski closure of $\Gamma _1$ is $H_1$. We will only consider $\Gamma _1$ because the exponential mixing of $\Gamma$ follows from the same statement for $\Gamma _1$ by taking covering space.

Let $H_o$ be the analytic connected component of $H_1$ containing identity. Since $\Gamma$ is non-elementary, the group $H_o$ doesn’t fix any point on the boundary. By a classic result (see [14] for example), up to conjugacy, $H_o$ preserves a hyperbolic subspace ${\mathbb {H}}^m$ with $m\le d$ and the restriction of $H_o$ to ${\mathbb {H}}^m$ contains ${\text {SO}}(m,1)^{\circ }$ with compact kernel. Preserving subspace is a Zariski closed condition, we know that $H_1$ also preserves ${\mathbb {H}}^m$ and the restriction of $H_1$ to ${\mathbb {H}}^m$ satisfies the same properties as $H_o$. Since $\Gamma _1$ is a torsion free discrete subgroup, the restriction map $\Gamma _1\rightarrow \Gamma _1|_{{\mathbb {H}}^m}$ is injective. Then the Zariski closure of $\Gamma _1|_{{\mathbb {H}}^m}$ also contains ${\text {SO}}(m,1)^{\circ }$. At most passing to an index 4 subgroup, we can suppose that $\Gamma _1|_{{\mathbb {H}}^m}$ is a subgroup of ${\text {SO}}(m,1)^\circ$. Hence $\Gamma _1|_{{\mathbb {H}}^m}$ is Zariski dense in ${\text {SO}}(m,1)^{\circ }$ and geometrically finite. (Definition 2.2 (2) implies that $\Gamma _1|_{{\mathbb {H}}^m}$ is still geometrically finite.) The BMS measure $m^{{\text {BMS}}}$ of $\Gamma _1$ on the unit tangent bundle $\Gamma _1\backslash {\text {T}}^1{\mathbb {H}}^{d+1}$ is actually supported on $\Gamma _1\backslash {\text {T}}^1{\mathbb {H}}^m$, which is the Zariski dense case. $\square$

4 The geodesic flow and the boundary map

For the rest of the paper, our standing assumption is

$$\begin{aligned}&\Gamma <G\,\,\,\text {Zariski dense, torsion-free, geometrically finite with parabolic elements}\\&\text {and}\,\,\,\infty \,\,\,\text {is a parabolic fixed point of}\,\,\, \Gamma . \end{aligned}$$

Let $\Delta _0:=\Delta _\infty$ be a fundamental region for the parabolic fixed point $\infty$ described in Sect. 2.3. In Sect. 6, we will construct a coding of the limit set satisfying the following properties.

Proposition 4.1

There are constants $C_1>0$, $\lambda ,\,\epsilon _0\in (0,1)$, a countable collection of disjoint, open subsets $\{\Delta _j\}_{j\in {\mathbb {N}}}$ in $\Delta _0$ and an expanding map $T:\sqcup _j\Delta _j\rightarrow \Delta _0$ such that:

• 1. $\sum _{j}\mu (\Delta _j)=\mu (\Delta _0)$.

• 2. For each j, there is an element $\gamma _j\in \Gamma$ such that $\Delta _j=\gamma _j \Delta _0$ and $T|_{\Delta _j}=\gamma _j^{-1}$.

• 3. Each $\gamma _j$ is a uniform contraction: $|\gamma _j'(x)|\le \lambda$ for all $x\in \Delta _0$.

• 4. For each $\gamma _j$, $|\mathrm D(\log |\gamma _j'|)(x)|<C_1$ for all $x\in \Delta _0$, where $\mathrm D(\log |\gamma _j'|)(x)$ is the differential of the map $z\mapsto \log |\gamma _j'(z)|$ at x.

• 5. Let R be the function on $\sqcup _j \Delta _j$ given by $R(x)=\log |\mathrm DT(x)|$. Then $\int e^{\epsilon _o R}\mathrm {d}\mu <\infty .$

Denote by ${\mathcal {H}}=\{\gamma _j\}_{j\in {\mathbb {N}}}$ the set of inverse branches of T. The last property is known as the exponential tail property and we will prove another form instead:

$$\begin{aligned} \sum _{\gamma \in {{\mathcal {H}}} }|\gamma '|_\infty ^{\delta -\epsilon _o}<\infty , \end{aligned}$$

(4.2)

where $|\gamma '|_\infty =\sup _{x\in \Delta _0}|\gamma '(x)|$. Proposition 4.1 (5) can be deduced from (4.2) by separating the integral to the sum of integrals over $\Delta _j$ and using quasi-invariance of PS measure.

Using Proposition 4.1, it can be shown that there exists a T-invariant ergodic probability measure $\nu$ on $\Delta _0$ which is absolutely continuous with respect to PS measure and the density function ${\bar{f}}_0$ is a positive Lipschitz function bounded away from 0 and $\infty$ on $\Delta _0\cap \Lambda _\Gamma$ (see for example [51, Lemma 2]).

The coding satisfies uniform nonintegrable condition (UNI). Let

$$\begin{aligned}&R_n(x):=\sum _{0\le k\le n-1}R(T^k(x))\,\,\,\text {for}\,\,\,x\,\,\,\text {with}\,\,\,T^k(x)\in \sqcup _j \Delta _j\\&\quad \qquad \text {for all}\,\,\,0\le k\le n-1,\\&{\mathcal {H}}^n=\{\gamma _{j_1} \cdots \gamma _{j_n}:\,\gamma _{j_k}\in {\mathcal {H}}\,\,\,\text {for}\,\,\,1\le k\le n\}. \end{aligned}$$

For $\gamma \in {{\mathcal {H}}}^n$, we have $R_n(\gamma x)=-\log |\gamma '(x)|$. Set

$$\begin{aligned} C_2=C_1/(1-\lambda ). \end{aligned}$$

(4.3)

Then by Proposition 4.1 (3) and (4), we obtain for any $\gamma \in {\mathcal {H}}_n$,

$$\begin{aligned} \sup _{x\in \Delta _0}\left| \mathrm D(\log |\gamma '|)(x)\right| \le C_2. \end{aligned}$$

(4.4)

Lemma 4.5

(UNI) There exist $r>0$ and $\epsilon _0>0$ such that for any $C>1$ the following holds for any large $n_0$. There exist $j_0\in {\mathbb {N}}$ and $\{\gamma _{mj}:1\le m\le 2, 1\le j\le j_0\}$ in ${\mathcal {H}}_{n_0}$ such that for any $x\in \Lambda _\Gamma \cap {\overline{\Delta }}_0$ and any unit vector $e\in {\mathbb {R}}^d$ there exists $j\le j_0$ such that for all $y\in B(x,r)$

$$\begin{aligned} |\partial _e(\tau _{1j}-\tau _{2j})(y)|\ge \epsilon _0, \end{aligned}$$

(4.6)

where $\tau _{mj}(x)=R_{n_0}(\gamma _{mj}x)$. Moreover, for all m, j,

$$\begin{aligned} |\mathrm D\tau _{mj}|_\infty \le C_2, \,\,\, |\gamma _{mj}'|_\infty \le \epsilon _0/C. \end{aligned}$$

(4.7)

The expanding map in the coding gives a contracting action in a neighborhood of $\infty$.

Lemma 4.8

There exist $0<\lambda <1$ and a neighbourhood $\Lambda _-$ of $\infty$ in $\Lambda _{\Gamma }$ such that $\Lambda _{-}$ is disjoint from ${\overline{\Delta }}_0$ and for any $\gamma \in {{\mathcal {H}}}$ and any $y,y'\in \Lambda _-$,

$$\begin{aligned} \gamma ^{-1}(\Lambda _{-})\subset \Lambda _{-},\ \ d_{{\mathbb {S}}^d}(\gamma ^{-1}y,\gamma ^{-1}y')\le \lambda d_{{\mathbb {S}}^d}(y,y'). \end{aligned}$$

(4.9)

The proofs of these results will be postponed to Sect. 6. Proposition 4.1 and Lemma 4.8 will be proved at the end of Sect. 6.8 and Lemma 4.5 will be proved in Sect. 6.9.

4.1 A semiflow over hyperbolic skew product

Hyperbolic skew product. We construct a hyperbolic skew product using Lemma 4.8. Let $\Lambda _{+}=\Lambda _{\Gamma }\cap \left( \sqcup _j\Delta _j\right)$ and $\Lambda _{-}$ be given as Lemma 4.8. Define the map ${\hat{T}}$ on $\Lambda _+\times \Lambda _-$ by

$$\begin{aligned} {\hat{T}}(x,x_-)=(\gamma _j^{-1}x,\gamma _j^{-1}x_-)\,\,\,\text {for}\,\,\, (x,x_-)\in \Lambda _+\times \Lambda _- \,\,\,\text {with}\,\,\, x\in \Delta _j, \end{aligned}$$

(4.10)

where $\gamma _j$ is given as in Proposition 4.1 (2). Lemma 4.8 implies $\gamma ^{-1}\Lambda _-\subset \Lambda _-$ for any $\gamma \in {\mathcal {H}}$. So ${\hat{T}}$ is well-defined.

Let $p:\Lambda _+\times \Lambda _-\rightarrow \Lambda _+$ be the projection to the first coordinate. This gives rise to a semiconjugacy between ${\hat{T}}$ and T. We equip $\Lambda _+\times \Lambda _-$ with the metric

$$\begin{aligned} d((x,x_-),(x',x_-'))=d_E(x,x')+d_{{\mathbb {S}}^d}(x_-,x_-'). \end{aligned}$$

(4.9) implies that the action of ${\hat{T}}$ on the fibre $\{ x\}\times \Lambda _-$ is contracting. Using this observation,we obtain

Proposition 4.11

• 1. There exists a unique ${\hat{T}}$-invariant, ergodic probability measure ${\hat{\nu }}$ on $\Lambda _+\times \Lambda _-$ whose projection to $\Lambda _+$ is $\nu$.

• 2. We have a disintegration of $\hat{\nu }$ over $\nu$: for any continuous function w on $\Lambda _+\times \Lambda _-$,

  $$\begin{aligned} \int _{\Lambda _+\times \Lambda _-} w\mathrm {d}\hat{\nu }=\int _{\Lambda _+}\int _{\Lambda _-}w\mathrm {d}\nu _x(x_-)\mathrm {d}\nu (x). \end{aligned}$$

  Moreover, there exists $C>0$ such that for any Lipschitz function w on $\Lambda _+\times \Lambda _-$, defining ${\bar{w}}(x)=\int w\mathrm {d}\nu _x$, we have

  $$\begin{aligned} \Vert {\bar{w}}\Vert _{\mathrm{Lip}}\le C\Vert w\Vert _{\mathrm{Lip}}. \end{aligned}$$

Proof

For the first statement, see [27, Theorem A] or [10, Proposition 1]. For the second statement, see [10, Proposition 3, Proposition 6], where they consider Riemannian manifold case but the same proofs also work in our fractal case. $\square$

Remark

The measure ${\hat{\nu }}$ is actually independent of the choice of the stable direction $\Lambda _-$: any $\Lambda _-$ satisfying Lemma 4.8 will lead to the same measure $\hat{\nu }$.

Hyperbolic skew product flow. Let $R:\Lambda _+\rightarrow {\mathbb {R}}_+$ be the function given in Proposition 4.1. By abusing notation, define $R:\Lambda _+\times \Lambda _-\rightarrow {\mathbb {R}}_+$ by setting $R(x,x_-)=R(x)$. Define the space

$$\begin{aligned} \Lambda ^{R}=\{(x,x_-,s)\in \Lambda _+\times \Lambda _-\times {\mathbb {R}}:\,0\le s< R(x,x_-)\}. \end{aligned}$$

Let $R_n=\sum _{j=0}^{n-1}R\circ {\hat{T}}^j$. The hyperbolic skew product flow $\{{\hat{T}}_t\}_{t\ge 0}$ over $\Lambda ^R$ is defined by ${\hat{T}}_t(x,x_-,s)=({\hat{T}}^n(x,x_-),s+t-R_n(x,x_-))$ for $\hat{\nu }$-almost every x, where n is the nonnegative integer such that $0\le s+t-R_n(x,x_-)<R({\hat{T}}^n(x,x_-))$. We equip $\Lambda ^R$ with the measure $\mathrm {d}\hat{\nu }^R:=\mathrm {d}\hat{\nu }\times \mathrm {d}t/{\bar{R}}$, where dt is Lebesgue measure on ${\mathbb {R}}_+$ and ${\bar{R}}=\int _{\Lambda _+\times \Lambda _-}R\mathrm {d}\hat{\nu }$. This is a ${\hat{T}}_t$-invariant ergodic measure.

Remark

We don’t use the commonly used “suspension space” construction to construct $\Lambda ^R$. The reason is that we will use a cutoff function in the proof of Theorem 1.1 and such cutoff functions are ill-defined in the suspension space, which is a quotient space of $\Lambda _+\times \Lambda _-\times {\mathbb {R}}$.

For any $L^{\infty }$ function $w:\Lambda ^{R}\rightarrow {\mathbb {R}}$, the Lipschitz norm of w is defined by

$$\begin{aligned} \Vert w\Vert _{{\text {Lip}}}=|w|_\infty +\sup _{ (y,a)\ne (y',a')\in \Lambda ^R}\frac{|w(y,a)-w(y',a')|}{d(y,y')+|a-a'|}. \end{aligned}$$

(4.12)

In Sect. 8, we will prove that ${\hat{T}}_t$ is exponential mixing with respect to $\hat{\nu }^R$.

Theorem 4.13

There exist $\epsilon _1>0$ and $C>1$ such that for any Lipschitz functions u, w on $\Lambda ^R$ and any $t>0$, we have

$$\begin{aligned} \left| \int u\ w\circ {\hat{T}}_t\mathrm {d}{\hat{\nu }}^R-\int u \mathrm {d}{\hat{\nu }}^R\int w\mathrm {d}{\hat{\nu }}^R\right| \le Ce^{-\epsilon _1 t}\Vert u\Vert _{{\text {Lip}}}\Vert w\Vert _{{\text {Lip}}}. \end{aligned}$$

4.2 Exponential mixing of geodesic flow

The map from $\Lambda ^R$ to ${\text {T}}^1(M)$. We construct a map from $\Lambda ^R$ to ${\text {T}}^1(M)$ which allows us to deduce the exponential mixing of the geodesic flow from that of ${\hat{T}}_t$.

Recall the Hopf parametrization in Sect. 2.4. We introduce the following time change map to have the function R given by derivative (see Proposition 4.1):

$$\begin{aligned} {\tilde{\Phi }}:\Lambda ^R\rightarrow \partial ^2({\mathbb {H}}^{d+1})\times {\mathbb {R}},\,\,\, (x,x_-,s)\mapsto (x,x_-,s-\log (1+|x|^2)). \end{aligned}$$

The map ${\tilde{\Phi }}$ induces a map $\Phi :\Lambda ^R\rightarrow {\text {T}}^1(M)$, where we use the Hopf parametrization to identify ${\text {T}}^1(M)$ with $\Gamma \backslash \partial ^2({\mathbb {H}}^{d+1})\times {\mathbb {R}}$. Note that $\Lambda _+\times \{\infty \}\times \{0\}$ is mapped to the unstable horosphere based at $\infty$ and passing o. The map $\Phi$ defines a semiconjugacy between two flows:

$$\begin{aligned} \Phi \circ {\hat{T}}_t={\mathcal {G}}_t\circ \Phi ,\,\,\,\text {for}\,\,\,t\ge 0. \end{aligned}$$

(4.14)

To see this, note that for any $(x,x_-,s)\in \Lambda ^R$, we have the expresssion

$$\begin{aligned} {\hat{T}}_t(x,x_-,s)&=({\hat{T}}^n(x,x_-),s+t-R_n(x,x_-)),\\ {\hat{T}}^n(x,x_-)&=\gamma ^{-1}(x,x_-)\,\,\,\text {for some}\,\,\,\gamma \in {{\mathcal {H}}}^n. \end{aligned}$$

By straightforward computation, we obtain

$$\begin{aligned} {\tilde{\Phi }}\circ {\hat{T}}_t(x,x_-,s)={\mathcal {G}}_t\circ \gamma ^{-1}{\tilde{\Phi }}(x,x_-,s), \end{aligned}$$

which leads to (4.14) by passing to the quotient space.

Relating $\hat{\nu }^{R}$ with $m^{{\text {BMS}}}$. The map $\Phi$ is not injective in general. Nevertheless, we are able to use $(\Lambda ^R,{\hat{T}}_t,\hat{\nu }^R)$ to study $({\text {T}}^1(M), {\mathcal {G}}_t,m^{{\text {BMS}}})$. The main result is the following proposition.

Proposition 4.15

The map $\Phi :(\Lambda ^R,{\hat{T}}_t,\hat{\nu }^R)\rightarrow ({\text {T}}^1(M),{\mathcal {G}}_t,m^{{\text {BMS}}})$ is a factor map, i.e.,

$$\begin{aligned} \Phi _*\hat{\nu }^R=m^{{\text {BMS}}}\,\,\, \text {and}\,\,\,\Phi \circ {\hat{T}}_t={\mathcal {G}}_t\circ \Phi \,\,\,\text {for all}\,\,\,t\ge 0. \end{aligned}$$

We need two lemmas to prove this proposition.

Lemma 4.16

There exists a measurable subset U in $\Lambda ^R$ such that by setting $V=\Phi (U)$ in ${\text {T}}^1(M)$, the restriction map of $\Phi$ on U gives a bijection between U and V. Moreover, the set V is of positive BMS measure.

Proof

We make use of the following commutative diagram

where $\pi$ is the covering map. Let $\epsilon >0$ be a number such that $\epsilon <\inf _{(x,x_-)\in \Lambda _+\times \Lambda _-}R(x,x_-)$. Set $S=\Lambda _+\times \Lambda _-\times [0,\epsilon )$. The restriction map ${\tilde{\Phi }}|_S$ gives a bijection between S and its image. Pick any $x\in S$. As $\pi$ is a covering map, there exists an open set $W\subset \partial ^2({\mathbb {H}}^{d+1})\times {\mathbb {R}}$ containing ${\tilde{\Phi }}(x)$ such that the restriction map $\pi |_W$ is a bijection. The sets $U={\tilde{\Phi }}^{-1}(W\cap {\tilde{\Phi }}(S))$ in $\Lambda ^R$ and $V=\pi (W\cap {\tilde{\Phi }}(S))$ satisfy the proposition. $\square$

Lemma 4.17

Let ${{\mathcal {Q}}}'$ be any subset in $\Lambda ^R$ with full $\hat{\nu }^R$ measure and ${{\mathcal {Q}}}$ be any subset in ${\text {T}}^1 (M)$ with full $m^{{\text {BMS}}}$ measure. Then there exist $x\in {{\mathcal {Q}}}'$ and $y\in {{\mathcal {Q}}}$ such that $\Phi (x)$ and y are in the same stable leaf.

Proof

The idea of the proof is straightforward: we make use of the local product description of $\hat{\nu }^R$ and $m^{{\text {BMS}}}$.

Let $\Phi _U$ be the restriction of $\Phi$ on U. In view of Lemma 4.16, we can consider the measure $\Phi _U^*(m^{{\text {BMS}}}|_V)$ on U, the pull back of $m^{{\text {BMS}}}|_V$, and denote it by m for simplicity. We can choose U and V sufficiently small so that m is given by

$$\begin{aligned} \mathrm {d}m(x,x_-,t)=cD(x,x_-)^{-2\delta }\mathrm {d}\mu (x)\mathrm {d}\mu (x_-)dt, \end{aligned}$$

where c is a positive constant and $D(x,x_-)=e^{\beta _{x}(o,x_*)/2} e^{\beta _{x_-}(o,x_*)/2}$ known as the visual distance. Let $p:\Lambda ^R\rightarrow \Lambda _+\times {\mathbb {R}}$ be the projection map, forgetting the $\Lambda _-$-coordinate. Then the pushforward measure $p_*m$ is given by

$$\begin{aligned} \mathrm {d}p_*m(x,t)=c\mathrm {d}\mu (x)\mathrm {d}t\int _{\{x\}\times \Lambda _-\times \{t\}\cap U} D(x,x_-)^{-2\delta }\mathrm {d}\mu (x_-). \end{aligned}$$

So it is absolutely continuous with respect to the measure $\mathrm {d}\nu \otimes \mathrm {d}t$.

We can find a set of the form $B=B_+\times \Lambda _-\times (t_1,t_2)$ such that $\nu (B_+)>0$ and $m(B\cap U)>0$. The pushforward measure $p_*(\hat{\nu }^R|_B)$ is given by

$$\begin{aligned} dp_*(\hat{\nu }^R|_B)=\mathrm {d}\nu \otimes \mathrm {d}t. \end{aligned}$$

(4.18)

On the one hand, we have that $p({{{\mathcal {Q}}}}'\cap B)$ is a conull set in p(B) with respect to $p_*(\hat{\nu }^R|_B)$.

On the other hand, we consider $B':=\Phi _U^{-1}({{{\mathcal {Q}}}}\cap V)\cap B$. It is a set with positive m measure and hence $p_*(B')$ is of positive $p_*(m)$ measure. The fact that $dp_*(m)$ is absolutely continuous with respect to $\mathrm {d}\nu \otimes \mathrm {d}t$ and (4.18) imply that $p_*(B')$ is of positive $p_*(\hat{\nu }^R|_B)$ measure. Therefore,

$$\begin{aligned} p({{{\mathcal {Q}}}}'\cap B)\cap p(B')\ne \emptyset . \end{aligned}$$

Let (x, t) be a point in the intersection. Then the points $(x,x_-,t)\in {{\mathcal {Q}}}'$ and $\Phi (x,x_-',t)\in {{\mathcal {Q}}}$ satisfy the conditions of the lemma. $\square$

Proof of Proposition 4.15

Let f be a $C^1$ function on ${\text {T}}^1(M)$ with finite $C^1$-norm. Since $m^{{\text {BMS}}}$ is ergodic [49], by Birkhoff ergodic theorem, for $m^{{\text {BMS}}}$-a.e. y in ${\text {T}}^1(M)$

$$\begin{aligned} \lim _{T\rightarrow +\infty }\frac{1}{T}\int _{0\le t\le T} f({{\mathcal {G}}}_t y)\mathrm {d}t=\int f\mathrm {d}m^{{\text {BMS}}}. \end{aligned}$$

(4.19)

Let ${{\mathcal {Q}}}$ be the set of points at which (4.19) hold and it is a set of full $m^{{\text {BMS}}}$ measure.

We consider $f\circ \Phi$, which can be thought as the lifting of f to $\Lambda ^R$. It is $\hat{\nu }^R$-integrable. Since ${\hat{T}}_t$ is mixing with respect to $\hat{\nu }^R$, by Birkhoff ergodic theorem, for $\hat{\nu }^R$-a.e. x,

$$\begin{aligned} \lim _{T\rightarrow +\infty }\frac{1}{T}\int _{0\le t\le T} f\circ \Phi ({\hat{T}}_t x)\mathrm {d}t=\int f\circ \Phi \mathrm {d}\hat{\nu }^R. \end{aligned}$$

Using the semiconjugacy $\Phi \circ {\hat{T}}_t={{\mathcal {G}}}_t\circ \Phi$, we actually have

$$\begin{aligned} \lim _{T\rightarrow +\infty }\frac{1}{T}\int _{0\le t\le T} f({{\mathcal {G}}}_t\Phi x)\mathrm {d}t=\int f\circ \Phi \mathrm {d}\hat{\nu }^R. \end{aligned}$$

(4.20)

Let ${{\mathcal {Q}}}'$ be the set of points at which (4.20) hold and it is a set of full $\hat{\nu }^R$ measure.

By Lemma 4.17, there exist points $x\in \Lambda ^R$ and $y\in {\text {T}}^1(M)$ such that $\Phi (x)$ and y are in the same stable leaf. Due to $d({{\mathcal {G}}}_t y,{{\mathcal {G}}}_t\Phi x)\rightarrow 0$ as $t\rightarrow +\infty$ and the uniform continuity of f,

$$\begin{aligned} \lim _{T\rightarrow +\infty }\left( \frac{1}{T}\int _{0\le t\le T} f({{\mathcal {G}}}_t y)\mathrm {d}t-\frac{1}{T}\int _{0\le t\le T} f({{\mathcal {G}}}_t\Phi x)\mathrm {d}t\right) =0. \end{aligned}$$

Therefore, we can deduce that

$$\begin{aligned} \int f\mathrm {d}m^{{\text {BMS}}}=\int f\circ \Phi \mathrm {d}\hat{\nu }^R. \end{aligned}$$

The above equation holds for every $C^1$ function on ${\text {T}}^1(M)$. The proof is complete. $\square$

Proof of Theorem 3.1. We are ready to prove Theorem 3.1. With Theorem 4.13 and Proposition 4.15 available, the work lies in the comparing the norm of the functions on $\Lambda ^{R}$ with that on ${\text {T}}^1(M)$. This is not obvious. Consider two points of the form (y, a) and $(y',a)$ in $\Lambda ^R$. By (4.12), $d((y,a),(y',a))$ remains the same when a changes. But if these two points are projected to ${\text {T}}^1(M)$, changing a means flowing these two points by the geodesic flow and $d(\Phi ((y,a),\Phi (y',a)))$ will change. Moreover, the function R used to define $\Lambda ^R$ is unbounded, making the argument more complex.

Proof of Theorem 3.1

Let u, v be any two $C^1$-functions on ${\text {T}}^1(M)$ with finite $C^1$-norm. Without loss of generality, we may assume that $m^{{\text {BMS}}}(u)=0$. Set $U=u\circ \Phi$ and $W=w\circ \Phi$. Using the semiconjugacy of $\Phi$, we obtain

$$\begin{aligned} \int u\cdot w\circ {\mathcal {G}}_t \mathrm {d}m^{{\text {BMS}}}=\int U \cdot W\circ {\hat{T}}_t\mathrm {d}\hat{\nu }^R. \end{aligned}$$

We use a cutoff function to relate the norms of U, W with those of u, w. Let $\epsilon >0$ be a constant less than $\epsilon _1/2$. Let $\tau _t$ be a decreasing Lipschitz function on $[0,\infty )$ such that $\tau _t=1$ on $[0,\epsilon t]$, $\tau _t =0$ on $(\epsilon t+1,\infty )$ and $|\tau _t|_{{\text {Lip}}}<2$. Set $U_t=U\cdot \tau _t$ and $W_t=W\cdot \tau _t$. For any two points (y, a) and $(y',a')$ (we may assume $a\ge a'$), we have

$$\begin{aligned}&|U_t(y,a)-U_t(y',a')|\\&\quad \le |U_t(y,a)-U_t(y,a')|+|U_t(y,a')-U_t(y',a')|\\&\quad \le \tau _t(a')|U(y,a)-U(y,a')|+|u|_{\infty } |\tau _t(a)-\tau _t (a')|\\&\qquad +\tau _t(a')|U(y,a')-U(y',a')|\\&\quad \ll |u|_{C^1}|a-a'|+|u|_{\infty } |a-a'|+e^{\epsilon t} |u|_{C^1} d(y,y'), \end{aligned}$$

where to obtain the last inequality, we use the fact $d(\Phi (y,a'),\Phi (y',a'))\le e^{a'}d(y,y')$ and $\tau _t\ne 0$ only on $[0,\epsilon t+1]$. Therefore, we have

$$\begin{aligned} \Vert U_t\Vert _{{\text {Lip}}}\ll e^{\epsilon t} \Vert u\Vert _{C^1}. \end{aligned}$$

(4.21)

A verbatim of the above argument also implies $\Vert W_t\Vert _{{\text {Lip}}}\ll e^{\epsilon t} \Vert w\Vert _{C^1}$.

We also need the following $L^1$-estimate. Using the exponential tail condition (Proposition 4.1 (5)), we obtain

$$\begin{aligned}&|U_t-U|_{L^1(\hat{\nu }^R)}\le |u|_{\infty } \int \max \{R(x)-\epsilon t,0\}\mathrm {d}\nu (x) \nonumber \\&\quad \ll |u|_{\infty } \int e^{\epsilon _o (R(x)-\epsilon t)}\mathrm {d}\nu (x)\ll e^{-\epsilon _o \epsilon t}|u|_{\infty }. \end{aligned}$$

(4.22)

The similar estimate holds for $W_t-W$. As $m^{{\text {BMS}}}(u)=0$, we have

$$\begin{aligned} |\int U_t\mathrm {d}\hat{\nu }^R|\ll e^{-\epsilon _o \epsilon t}|u|_{\infty }. \end{aligned}$$

(4.23)

Using Theorem 4.13 together with (4.21), (4.22) and (4.23), we obtain

$$\begin{aligned}&|\int U \cdot W\circ {\hat{T}}_t\mathrm {d}\hat{\nu }^R|\\&\quad \le |\int U_t\cdot W_t\circ {\hat{T}}_t\mathrm {d}\hat{\nu }^R|\\&\qquad +|\int (U-U_t)\cdot W_t\circ {\hat{T}}_t\mathrm {d}\hat{\nu }^R|+|\int U\cdot (W-W_t)\circ {\hat{T}}_t\mathrm {d}\hat{\nu }^R |\\&\quad \ll |\int U_t\mathrm {d}\hat{\nu }^R|\cdot |\int W_t\mathrm {d}\hat{\nu }^R|\\&\qquad +e^{-\epsilon _1 t}\Vert U_t\Vert _{{\text {Lip}}}\Vert W_t\Vert _{{\text {Lip}}}+|w|_\infty |U-U_t|_{L^1(\hat{\nu }^R)}\\&\qquad +|u|_\infty |W-W_t|_{L^1(\hat{\nu }^R)}\\&\quad \ll (e^{-(\epsilon _1-2\epsilon )t}+e^{-\epsilon _o\epsilon t} )|u|_{C^1}|w|_{C^1}. \end{aligned}$$

Due to $\epsilon <\epsilon _1/2$, the proof is complete. $\square$

5 Parabolic fixed points and measure estimate

In this section, we provide a detailed description of the $\Gamma$-action on $\partial {\mathbb {H}}^{d+1}$ and different types of estimate for the PS measure.

5.1 Explicit computation

Let $H_\infty$ be the horoball based at $\infty$ given by ${\mathbb {R}}^d\times \{x\in {\mathbb {R}}:\, x> 1\}$. For a horoball H, we define the height of the horoball by

$$\begin{aligned} h(H):=\sup _{y\in H}h(y). \end{aligned}$$

Lemma 5.1

Suppose $g\in G$ is not in $Stab_G(\infty )$. Let $p=g \infty$ and $p'=g ^{-1}\infty$. Then we have

• $$\begin{aligned} h(gH_\infty )=h(g^{-1}H_\infty ). \end{aligned}$$

  (5.2)

• For any $x\in {\mathbb {H}}^{d+1}\cup \partial {\mathbb {H}}^{d+1}$, we have

  $$\begin{aligned} g x= & {} h(gH_\infty ) \frac{x-(p',0)}{|x-(p',0)|^2}\begin{pmatrix} A &{} 0\\ 0 &{} 1\end{pmatrix}+(p,0) \nonumber \\ g ^{-1}x= & {} h(gH_\infty )\frac{x-(p,0)}{|x-(p,0)|^2}\begin{pmatrix}A^{-1} &{} 0\\ 0 &{} 1\end{pmatrix}+(p',0), \end{aligned}$$

  (5.3)

  where $A\in {\text {SO}}(d)$, and we view x, (p, 0) and $(p',0)$ as row vectors in ${\mathbb {R}}^{d+1}$.

Proof

By Proposition A.3.9 (2) in [12], the action of g on the upper half space is given by

$$\begin{aligned} g x=\lambda \iota (x) \begin{pmatrix} A &{} 0\\ 0 &{}1 \end{pmatrix}+(b,0), \end{aligned}$$

where A is in ${\text {SO}}(d)$, $\lambda \in {\mathbb {R}}^+$, $b\in {\mathbb {R}}^d$ and $\iota (x)$ either equals x or is given by an inversion with respect to a unit sphere centered at ${\mathbb {R}}^d\times \{0\}$. In fact, this is the Bruhat decomposition of G. Since g does not fix $\infty$, $\iota (x)$ is an inversion. We have for any $x\in {\mathbb {H}}^{d+1}$

$$\begin{aligned} g x=\lambda \frac{x-(x',0)}{|x-(x',0)|^2} \begin{pmatrix} A &{} 0\\ 0 &{}1 \end{pmatrix}+(b,0) \end{aligned}$$

with $x'\in {\mathbb {R}}^d$. Hence $b=g \infty =p$ and $x'=g ^{-1}\infty =p'$.

Note that

$$\begin{aligned} h(g x)=\lambda h(x)/|x-(p',0)|^2. \end{aligned}$$

Since g maps the original horoball $H_\infty ={\mathbb {R}}^d\times \{ x> 1\}$ to the horoball $gH_\infty$ based at p, it follows from the above formula that

$$\begin{aligned} h(gH_\infty )=\sup _{x\in {\mathbb {R}}^d\times \{1\} }h(g x)=\lambda . \end{aligned}$$

To obtain the formula for $g ^{-1}x$, note that

$$\begin{aligned} x=g (g ^{-1}x)=h(gH_\infty ) \frac{g ^{-1} x-(p',0)}{|g ^{-1} x-(p',0)|^2}\begin{pmatrix} A &{} 0\\ 0 &{} 1\end{pmatrix}+(p,0). \end{aligned}$$

This yields

$$\begin{aligned} \frac{1}{h(gH_\infty )}\left( x-(p,0)\right) \begin{pmatrix} A^{-1} &{} 0\\ 0 &{} 1\end{pmatrix}+(p',0)=\frac{g ^{-1} x-(p',0)}{|g ^{-1} x-(p',0)|^2}+(p',0). \end{aligned}$$

Applying both sides the inversion with respect to the unit sphere centered at $(p',0)$ and using the fact that $A\in {\text {SO}}(d)$, we obtain the formula for $g ^{-1}x$. Meanwhile, we apply to $g ^{-1} x$ the argument used to get the formula for gx. Comparing the formulas given by these two methods, we have that $h(gH_\infty )=h(g^{-1}H_\infty )$. $\square$

Lemma 5.4

For a horoball H based at $p\ne \infty$ and $g\in G$ not in $Stab_G(\infty )$, we have

$$\begin{aligned}&h(g H)\ge \frac{h(g ^{-1}H_\infty ) h(H)}{d(g ^{-1}\infty ,p)^2+h(H)^2}, \end{aligned}$$

(5.5)

$$\begin{aligned}&h(g H)\le \frac{h(g ^{-1}H_\infty ) h(H)}{(d(g ^{-1}\infty ,p)-h(H)/2)^2}. \end{aligned}$$

(5.6)

Proof

Using (5.3), we obtain

$$\begin{aligned} h(g H)\ge h(g (p,h(H)))=\frac{h(g ^{-1}H_\infty )h(H)}{d(p,g ^{-1}\infty )^2+h(H)^2}. \end{aligned}$$

For (5.6), we have

$$\begin{aligned} h(g H)=\sup _{y\in \partial H}h(g y)=\sup _{y\in \partial H} \frac{h(g ^{-1}H_\infty ) h(y)}{|y-(g^{-1}\infty ,0)|^2}. \end{aligned}$$

Note that for every $y\in \partial H$, we have $|y-(g^{-1}\infty ,0)|^2\ge d_{E}(y',g^{-1}\infty )^2\ge (d(g^{-1}\infty ,p)-h(H)/2)^2$, where $y'$ is the projection of y to $\partial {\mathbb {H}}^{d+1}$ and this yields (5.6). $\square$

Let ${{\mathcal {P}}}$ be the set of parabolic fixed points in $\partial {\mathbb {H}}^{d+1}$. Two parabolic fixed points are called equivalent if they are in the same $\Gamma$-orbit. Let ${\mathbf {P}}$ be a complete set of inequivalent parabolic fixed points. As $\Gamma$ is geometrically finite, the set ${\mathbf {P}}$ is finite. Suppose that ${\mathbf {P}}=\{p_1,\ldots ,p_j \}$ for some $j\in {\mathbb {N}}$, a complete set of inequivalent parabolic fixed points and set $p_1=\infty$.

Fix a collection of pairwise disjoint horoballs based at parabolic fixed points as follows. Without loss of generality, we may assume that $H_{\infty }$ is a horocusp region for $\infty$. For the parabolic fixed point $\gamma \infty$, we attach the horoball $H_{\gamma \infty }:=\gamma H_{\infty }$ to it. For other parabolic fixed point $p_i$ in ${\mathbf {P}}$, we fix a horoball $H_{p_i}$ based at $p_i$ which is a horocusp region for $p_i$. For other parabolic fixed points $\gamma p_i$ in the $\Gamma$ orbit of $p_i$, we attach the horoball $H_{\gamma p_i}:=\gamma H_{p_i}$ to it. By Definition 2.2, we can choose the horoballs in such a way that they are pairwise disjoint. For $p\in {\mathcal {P}}$, we define the height function h(p) of p as the height of $H_p$ based at p, that is

$$\begin{aligned} h(p):=h(H_p). \end{aligned}$$

For $x\in \partial {\mathbb {H}}^{d+1}$ and $r>0$, set B(x, r) to be the ball centered at x of radius r in $\partial {\mathbb {H}}^{d+1}$ with respect to the Euclidean metric.

Lemma 5.7

(Explicit computation) Suppose $\gamma$ is not in $Stab_\infty (\Gamma )$. Then for any $r>0$ and $x\in \partial {\mathbb {H}}^{d+1}$,

• $\gamma ^{-1}B(p,r)=B(p',h(p)/r)^c$,

• $|(\gamma ^{-1})'(x)|=h(p)/d(x,p)^2,\,\,\, |\gamma '(x)|=h(p)/d(x,p')^2$,

where $p=\gamma \infty$ and $p'=\gamma ^{-1}\infty$.

Proof

The first equation follows from (5.3) easily.

In view of Lemma 5.1, the computation of the derivative of inversion maps gives the expression of $|\gamma '(x)|$ and $|(\gamma ^{-1})'(x)|$. $\square$

5.2 Multi-cusps

Recall that ${\mathbf {P}}=\{p_1,\ldots ,p_j \}$. For each $p_i$, we consider a coordinate change transformation: let $g_i$ be an element in G such that $g_ip_i=\infty$. This $g_i$ is not unique and we can choose a $g_i$ such that $g_iH_{p_i}=H_\infty ={\mathbb {R}}^d\times \{x> 1 \}$. We will frequently make use of the following commutative diagram:

On the right hand side of the diagram, the acting group is $g_i\Gamma g_i^{-1}$ and $\infty$ is a parabolic fixed point of the group.

Once $g_i$’s are fixed, we consider the action of $g_i\Gamma g_i^{-1}$ and its parabolic fixed points $g_i{{\mathcal {P}}}$. We think they are in i-th hyperbolic space. Set the horoball $H_{g_ip}(i)=g_iH_p$ for $p\in {{\mathcal {P}}}$ and define the height

$$\begin{aligned} h(g_i p,i):=h(H_{g_ip}(i)). \end{aligned}$$

If there is no confusion, we always abbreviate $H_{g_ip}(i)$ and $h(g_i p,i)$ to $H_{g_ip}$ and $h(g_i p)$.

The results in Sect. 2.3 hold for each $p_i$. We have the group $(g_i\Gamma g_i^{-1})_\infty$, which is a maximal normal abelian subgroup in ${\text {Stab}}_{g_i\Gamma g_i^{-1}}(\infty )$. Write

$$\begin{aligned} \Gamma _{p_i}=g_i^{-1}(g_i\Gamma g_i^{-1})_\infty g_i. \end{aligned}$$

(5.9)

Let $\Delta _{p_i}'$ be a fundamental region for the parabolic fixed point $\infty$ under the $g_i\Gamma g_i^{-1}$-action. We can choose $\Delta _{p_i}'$ in such a way that for $\Delta _{p_i}=g_i^{-1}\Delta _{p_i}'$

$$\begin{aligned} \{p_i,\ 1\le i\le j\}\cap (\cup _{1\le k\le j}{\overline{\Delta }}_{p_k})=\emptyset . \end{aligned}$$

(5.10)

This choice is possible because, for each $p_i$, we can find a $\Delta _{p_i}'$ such that $\Delta _{p_i}$ sufficiently close to $p_i$ in $\partial {\mathbb {H}}^{d+1}$ under the spherical metric. Set

$$\begin{aligned} \Delta =\cup _{1\le k\le j}\Delta _{p_k}. \end{aligned}$$

By (5.10), we have ${\overline{\Delta }}\cap \{\infty \}=\emptyset$. So the set ${\overline{\Delta }}$ is compact.

Consider any parabolic fixed point $p=\gamma p_i$ with $\gamma \in \Gamma$. We know that such $\gamma$ is not unique (any element in $\gamma {\text {Stab}}_{\Gamma }(p_i)$ also works) and we fix a choice of $\gamma$ such that $\gamma ^{-1}p_i\in {\overline{\Delta }}_{p_i}$. We call $\gamma$ the representation of p. Set

$$\begin{aligned} x_p:=\gamma ^{-1}p_i. \end{aligned}$$

Lemma 5.11

There exists $C>1$ such that for $1< i\le j$ and for any parabolic fixed point p in $\Delta$, we have

$$\begin{aligned} 1/C\le h(g_i p)/h(p)\le C. \end{aligned}$$

Proof

Consider the action of $g_i$ on $\partial {\mathbb {H}}^{d+1}$. Notice that $p_i=g_i^{-1}\infty$ and $h(p_i)=h(H_{p_i})=h(g_i^{-1}H_\infty )$. Applying (5.5) to the horoball $H_p$ based at p and the element $g_i$, we obtain

$$\begin{aligned} h(g_i p)=h(g_iH_{p})\ge \frac{h(p_i) h(p)}{d(p_i,p)^2+h(p)^2}. \end{aligned}$$

It follows from ${\overline{\Delta }}$ compact that $d(p_i,p)$ is bounded for $p\in \Delta$. Then $h(g_i p)\ge h(p)/C$.

For the other inequality, notice that $g_ip_1=g_i\infty$ and $h(g_i p_1)=h(g_i H_\infty )$. Applying (5.5) with with the horoball $H_{g_ip}$ based at $g_ip$ and $g_i^{-1}$ we have

$$\begin{aligned} h(p)=h(g_i^{-1}H_{g_ip})\ge \frac{h(g_i p_1)h(g_i p) }{d(g_ip_1,g_ip)^2+h(g_i p)^2 }. \end{aligned}$$

It follows from (5.10) that $g_i{\overline{\Delta }}\cap \{\infty \}=g_i{\overline{\Delta }}\cap \{g_ip_i \}=\emptyset$. So $g_i{\overline{\Delta }}$ is compact and $d(g_ip_1,g_ip)$ is bounded for $p\in \Delta$. Then $h(p)\ge h(g_i p)/C$. $\square$

Lemma 5.12

For $1< i\le j$, the map $g_i:\Delta \rightarrow g_i\Delta$ is bi-Lipschitz.

Proof

By (5.3), we have

$$\begin{aligned} d(g_ix,g_iy)=h(p_i)d\left( \frac{x-p_i}{|x-p_i|^2},\frac{y-p_i}{|y-p_i|^2}\right) \le C|x-y|=Cd(x,y), \end{aligned}$$

where the inequality due to (5.10).

For the other direction, we use (5.3) to obtain

$$\begin{aligned} d(g_i^{-1}x,g_i^{-1}y)= & {} h(p_i)d\left( \frac{x-g_ip_1}{|x-g_ip_1|^2},\frac{y-g_ip_1}{|y-g_ip_1|^2}\right) \\\le & {} C|x-y|=Cd(x,y), \end{aligned}$$

where the inequality is due to that $d(x,g_ip_1)$ is bounded below for $x\in g_i\Delta$ by (5.10).$\square$

Patterson–Sullivan measure under conjugation. In the presence of multi-cusps, we need to consider the Patterson–Sullivan measure for the conjugation of $\Gamma$. Recall that $\{\mu _{y}\}_{y\in {\mathbb {H}}^{d+1}}$ is the $\Gamma$-invariant conformal density of dimension $\delta$ and we denoted $\mu _o$ by $\mu$ for short. For each $g_i$ with $1<i\le j$, set $\Gamma _i=g_i\Gamma g_i^{-1}$. The limit set $\Lambda _{\Gamma _i}$ is $g_i\Lambda _{\Gamma }$ and the critical exponent of $\Gamma _i$ equals $\delta$. For every $y\in {\mathbb {H}}^{d+1}$, define the following measure

$$\begin{aligned} {\tilde{\mu }}_{y}:=(g_i)_{*}\mu _{g_i^{-1}y}, \end{aligned}$$

where $(g_i)_{*}\mu _{g_i^{-1}y}(E)=\mu _{g_i^{-1}y}(g_i^{-1}E)$ for any Borel subset E of $\partial {\mathbb {H}}^{d+1}$. It is easy to check that ${\tilde{\mu }}_{y}$ is supported on $\Lambda _{\Gamma _i}$ and for any $y,z\in {\mathbb {H}}^{d+1}$, $x\in \partial {\mathbb {H}}^{d+1}$ and $\gamma \in \Gamma _i$,

$$\begin{aligned} \frac{\mathrm {d}{\tilde{\mu }}_{y}}{\mathrm {d}{\tilde{\mu }}_{z}}(x)=e^{-\delta \beta _{x}(y,z)}\,\,\, \text {and}\,\,\,(\gamma )_{*}{\tilde{\mu }}_{y}={\tilde{\mu }}_{\gamma y}. \end{aligned}$$

It follows from the uniqueness of $\Gamma _{i}$-invariant conformal density that this construction gives exactly the $\Gamma _i$-invariant conformal density on $\Lambda _{\Gamma _i}$ of dimension $\delta$. In later sections, we will denote ${\tilde{\mu }}_o$ by $\mu _{\Gamma _i}$ and the above analysis yields that for any Borel subset E of $\partial {\mathbb {H}}^{d+1}$

$$\begin{aligned} e^{-\delta d(o,g_io)} \mu (g_i^{-1}E)\le \mu _{\Gamma _i}(E)\le e^{\delta d(o,g_io)} \mu (g_i^{-1}E). \end{aligned}$$

(5.13)

5.3 Doubling property of PS measure

We start with two results: Proposition 5.14 and Lemma 5.15, deduced from [46, Theorem 2]. They used spherical metric, but locally it is equivalent to euclidean metric.

Proposition 5.14

• (Doubling property) For every $C>1$, there exists $\epsilon <1$ such that for every $x\in \Lambda _\Gamma \cap \Delta$ and $1/C\ge r>0$,

  $$\begin{aligned} \mu (B(x,r))>\epsilon \mu (B(x,Cr)). \end{aligned}$$

• (Growth of measure) There exists $C_{3}>1$, such that for every $x\in \Lambda _\Gamma \cap \Delta$ and $r<1/C_{3}$,

  $$\begin{aligned} 2\mu (B(x,r))<\mu (B(x,C_{3}r)). \end{aligned}$$

Lemma 5.15

Let p be a parabolic fixed point in $\Delta$ of rank k. For $0<r\le h(p)$,

$$\begin{aligned} \mu (B(p,r))\approx r^{2\delta -k}h(p)^{k-\delta }. \end{aligned}$$

(5.16)

Lemma 5.17

For every $C>1$, there exists $C'>1$ such that for every parabolic fixed point $p=\gamma \infty \in \Delta$ with $\gamma$ the representation, for any Borel subset $E\subset B(p, C h(p))$, we have

$$\begin{aligned} h(p)^{\delta }\mu (\gamma ^{-1}E)/C'\le \mu (E)\le C' h(p)^{\delta }\mu (\gamma ^{-1}E). \end{aligned}$$

Proof

As the PS measure is quasi-invariant, we have

$$\begin{aligned} \mu (\gamma ^{-1}E)=\int _{x\in E} |(\gamma ^{-1})' x|^{\delta } \left( \frac{1+| x|^2}{1+|\gamma ^{-1} x|^2}\right) ^{\delta }\mathrm {d}\mu (x). \end{aligned}$$

By Lemma 5.7, we have $|(\gamma ^{-1})' x|=h(p)/d(x,p)^2$. We also have

$$\begin{aligned} |d(\gamma ^{-1}\infty ,0)-h(p)/d(x,p)| \le |\gamma ^{-1} x|\le d(\gamma ^{-1}\infty ,0)+h(p)/d(x,p). \end{aligned}$$

Due to $\gamma ^{-1}\infty \in {\overline{\Delta }}$, if $h(p)/d(x,p)>\max \{2d(\gamma ^{-1}\infty ,0),1 \}$, then $1+|\gamma ^{-1}x|^2\approx (h(p)/d(x,p))^2$. Otherwise, due to $h(p)/d(x,p)\ge 1/C$, we also have $1+|\gamma ^{-1}x|^2\approx (h(p)/d(x,p))^2$. The lemma follows from these bounds on $|(\gamma ^{-1})' x|$ and $1+|\gamma ^{-1}x|^2$. The computation also makes sense even if $x=p$, because the ratio $|(\gamma ^{-1})'x|/(1+|\gamma ^{-1}x|^2)$ is always bounded in $B(p,Ch(p))-\{p\}$ and we can extend it continous to p. $\square$

Recall $\Delta _{0}:=\Delta _{\infty }=\Delta _{p_1}$.

Lemma 5.18

There exists $C>0$ such that for any Borel set E of diameter less than the diameter of $\Delta _{0}$ and $\gamma _1\in \Gamma _{\infty }$, we have that for any $x\in E$

$$\begin{aligned} \frac{\mu (\gamma _1E)}{\mu (E)}\in (1/C,C) \left( \frac{1+| x|^2}{1+|\gamma _1 x|^2}\right) ^{\delta }. \end{aligned}$$

Proof

Due to the derivative of $\gamma _1$ and the quasi-invariance of PS measure, we obtain

$$\begin{aligned} \mu (\gamma _1 E)=\int _{E}\left( \frac{1+| x|^2}{1+|\gamma _1 x|^2}\right) ^{\delta }\mathrm {d}\mu (x). \end{aligned}$$

Now for any x, y in E, we have

$$\begin{aligned} \frac{1+|x|^2}{1+|y|^2}=1+\frac{(|x|-|y|)(|x|+|y|)}{1+|y|^2}\le C', \end{aligned}$$

(5.19)

with $C'>1$ only depending on the diameter of E. The same argument also gives the same upper bound for $(1+|y|^2)/(1+|x|^2)$. The set $\gamma _{1}E$ is a set with the same diameter as E. So we also have

$$\begin{aligned} \frac{1+|\gamma _1x|^2}{1+|\gamma _1y|^2}\in (1/C',C') \end{aligned}$$

(5.20)

for any x, y in E. The proof is complete by applying (5.19) and (5.20) to the formula of $\mu (\gamma _1 E)$. $\square$

Lemma 5.21

There exist constants $c>0$ and $C>1$ such that for every parabolic fixed point $p\ne \infty$, if $r\le h(p)/C$, then

$$\begin{aligned} \mu (B(p,r)-B(p,r/\sqrt{e}))\ge c\mu (B(p,r)). \end{aligned}$$

Proof

Consider $p\in {\overline{\Delta }}_{0}$. Indeed, since $\Gamma _\infty {\overline{\Delta }}_{0}$ covers the intersection ${\mathbb {R}}^d\cap \Lambda _{\Gamma }$, we can always find a $\gamma _1$ in $\Gamma _{\infty }$ such that $\gamma _1 p\in \overline{\Delta _0}$. Then applying Lemma 5.18 to $E=B(p,r)$ and $E=B(p,r)-B(p,r/\sqrt{e})$, we have

$$\begin{aligned} \frac{\mu (B(p,r)-B(p,r/\sqrt{e}))}{\mu (B(p,r))}\approx \frac{\mu (B(\gamma _1p,r)-B(\gamma _1p,r/\sqrt{e}))}{\mu (B(\gamma _1p,r))}. \end{aligned}$$

We only need to give a lower bound to $\mu (B(p,r)-B(p,r/\sqrt{e}))$ and then use (5.16) to obtain Lemma 5.21.

Assume $p\in \overline{\Delta _0}$ is of rank k. Consider the case when $p=\gamma \infty$ with $\gamma \in \Gamma$ the representation of p. We claim that there exists a constant $C>1$ such that for $\gamma _1\in \Gamma _{\infty }$, with $\gamma \gamma _1\Delta _0\subset B(p,r)-B(p,r/\sqrt{e})$, we have

$$\begin{aligned} \mu (\gamma \gamma _1\Delta _0)\gg \frac{h(p)^{\delta }}{(d(\gamma _1\Delta _0,x_p)+C)^{2\delta }}. \end{aligned}$$

(5.22)

Proof of the claim: By Lemma 5.17, we have $\mu (\gamma \gamma _1\Delta _0)\approx h(p)^\delta \mu (\gamma _1\Delta _0)$. By Lemma 5.18, we have

$$\begin{aligned} \mu (\gamma _1\Delta _0)\approx \left( \frac{1+| x|^2}{1+|\gamma _1 x|^2}\right) ^{\delta }\mu (\Delta _0), \end{aligned}$$

for any $x\in \Delta _{0}$. Now, since $|\gamma _1x|\le d(\gamma _1\Delta _{0},x_p)+|x_p|+C'$, so

$$\begin{aligned} \mu (\gamma _1\Delta _0)\ge \mu (\Delta _{0}) /(d(\gamma _1\Delta _0,x_p)+C)^{2\delta } \end{aligned}$$

for some constant $C>1$.

By Lemma 5.7, we have $\gamma ^{-1}(B(p,r)-B(p,r/\sqrt{e}))=B(x_p,\sqrt{e}h(p)/r)-B(x_p,h(p)/r)$. Let $C'={\text {diam}}(\Delta _0)$. Let ${\mathbb {R}}^k$ be the subspace described in Lemma 2.3. For a set E in ${\mathbb {R}}^d$, we define ${\text {Vol}}_{{\mathbb {R}}^k}(E)$ as ${\text {Vol}}(E\cap {\mathbb {R}}^k)$. Since $x_p\in \Lambda _{\Gamma }\cap {\mathbb {R}}^d$ has bounded distance to ${\mathbb {R}}^k$, the number of $\gamma _1\Delta _0$’s in such region is at least

$$\begin{aligned}&{\text {Vol}}_{{\mathbb {R}}^k}\left( B\left( x_p,\sqrt{e}h(p)/r-C'\right) -B\left( x_p,h(p)/r+C'\right) \right) /{\text {Vol}}_{{\mathbb {R}}^k}(\Delta _0)\nonumber \\&\qquad \gg h(p)^kr^{-k}. \end{aligned}$$

(5.23)

Then (5.23) and (5.22) imply

$$\begin{aligned}&\mu (B(p,r)-B(p,r/\sqrt{e}))\\&\quad \ge \sum _{\gamma _1\Delta _0\subset B(x_p,\sqrt{e}h(p)/r)-B(x_p,h(p)/r)}\mu (\gamma \gamma _1\Delta _0)\\&\quad \gg h(p)^{k-\delta } r^{2\delta -k}. \end{aligned}$$

Consider general case when $p=\gamma p_i$ with $\gamma \in \Gamma$ the representation of p. We estimate the measure $\mu (\gamma \gamma _1 \Delta _{p_i})$ for any $\gamma _1\in \Gamma _{p_i}$ satisfying $\gamma \gamma _1\Delta _{p_i}\subset B(p,r)-B(p,r/\sqrt{e})$. Using  (5.13), we have

$$\begin{aligned} \mu (\gamma \gamma _1\Delta _{p_i})\approx \mu _{\Gamma _i} (g_i \gamma \gamma _1 \Delta _{p_i}), \end{aligned}$$

where $\Gamma _i:=g_i\Gamma g_i^{-1}$. Lemma 5.12 yields

$$\begin{aligned} g_i\gamma \gamma _1 \Delta _{p_i}\subset g_i(B(p,r)-B(p,r/\sqrt{e}))\subset B(g_ip, Cr)-B(g_ip,r/(C\sqrt{e})). \end{aligned}$$

(5.24)

So we can use the argument for the previous case to obtain

$$\begin{aligned} \mu _{\Gamma _i}(g_i\gamma \gamma _1\Delta _{p_i})\approx r^{2\delta } h(g_i p)^{-\delta }. \end{aligned}$$

(5.25)

Then we count the number of $\gamma \gamma _1\Delta _{p_i}$’s in $B(p,r)-B(p,r/\sqrt{e})$. It equals the number of $g_i\gamma \gamma _1\Delta _{p_i}$’s in $g_i(B(p,r)-B(p,r/\sqrt{e}))$. The map $g_i$ maps B(p, r) and $B(p,r/\sqrt{e})$ to two spheres and the distance between $g_i B(p,r)$ and $g_i B(p,r/\sqrt{e})$ is at least $(1-1/\sqrt{e})r/C$. The map $g_i\gamma ^{-1}g_i^{-1}$ maps $g_i B(p,r)$ and $g_i B(p,r/\sqrt{e})$ to two spheres and let R and $p'$ be the radius and the center of the outer sphere respectively. Using (5.24) and Lemma 5.7, we have

$$\begin{aligned} R\in (h(g_i p)/(Cr), C\sqrt{e}h(g_i p)/r). \end{aligned}$$

(5.26)

For every $x\in B(g_ip,Cr)-B(g_ip, r/(C\sqrt{e}))$, we have $|(g_i\gamma ^{-1}g_i^{-1})'(x)|\in (h(g_i p)/(C^2r^2), C^2eh(g_i p)/r^2)$. So the distance between $g_i\gamma ^{-1}B(p,r)$ and $g_i\gamma ^{-1}B(p,r/\sqrt{e})$ is at least $(1-1/\sqrt{e}) h(g_i p)/(C^3r)$. This distance estimate together with (5.26) implies there exists some constant $c\in (0,1)$ such that

$$\begin{aligned} g_i\gamma ^{-1} (B(p,r)-B(p,r/\sqrt{e}))\supset B(p', R)-B(p',cR). \end{aligned}$$

The number of $g_i\gamma _1\Delta _{p_i}$ in $g_i\gamma ^{-1}(B(p,r)-B(p,r/\sqrt{e}))$ is at least

$$\begin{aligned}&{\text {Vol}}_{{\mathbb {R}}^k}\left( B\left( p',R-C''\right) -B\left( p',cR+C''\right) \right) /{\text {Vol}}_{{\mathbb {R}}^k}(g_i\Delta _{p_i})\nonumber \\&\quad \gg R^k\gg h(g_i p)^k r^{-k}, \end{aligned}$$

(5.27)

where $C''={\text {diam}}(g_i\Delta _{p_i})$. A lower bound for $\mu (B(p,r)-B(p,r/\sqrt{e}))$ can be obtained using Lemma 5.11, (5.25) and (5.27). $\square$

5.4 Friendliness of PS measure

For any $r>0$, set

$$\begin{aligned} N_r(\Delta _0):=\{x\in \Delta _0:\,\ d(x,\partial \Delta _0)\le r \}. \end{aligned}$$

(5.28)

Lemma 5.29

There exist $0<\epsilon _0<1$ such that for all $0<\epsilon <\epsilon _0$ there exists $\lambda =\lambda (\epsilon )\in (0,1)$ for all $r<1$

$$\begin{aligned} \mu (N_{\epsilon r}(\Delta _0))\le \lambda \mu (N_r(\Delta _0)). \end{aligned}$$

(5.30)

Moreover, the constant $\lambda (\epsilon )$ tends to zero as $\epsilon$ tends to zero.

Recall from Sect. 2.3 that $\Delta _0=B_Y(C)\times \Delta _0'$. Recall that the set $\Delta _{0}'$ is a parallelotope. Let $l'$ be a facet of $\Delta _0'$ and $l=B_Y(C)\times l'$. Let $\gamma$ be the element in $\Gamma _{\infty }$ identifying $l'$ with the opposite facet $l''$ so $\gamma$ also identifies $B_Y(C)\times l'$ with $B_Y(C)\times l''$. Set

$$\begin{aligned} N_r(l):=\{x\in \Delta _0\cup \gamma ^{-1}\Delta _0:\,d(x,l)\le r\}. \end{aligned}$$

Lemma 5.29 is deduced from the following lemma.

Lemma 5.31

There exist $0<\epsilon ,\lambda <1$ such that for all $r<1$

$$\begin{aligned} \mu (N_{\epsilon r}(l))\le \lambda \mu (N_r(l)). \end{aligned}$$

Proof of Lemma 5.29

Assume that $\infty$ is a rank k cusp. If $\infty$ is not a cusp of maximal rank, then note that $(\partial B_Y(C))\times \Delta _0'=\{|y|=C\}\times \Delta _0'$ does not intersect $\Lambda _{\Gamma }$. A small neighborhood of this boundary has zero PS measure. Therefore, we just need to consider the neighborhood of l’s. Using Lemma 5.31, we obtain

$$\begin{aligned} \mu (N_{\epsilon r}(\Delta _0))\le \sum _{l}\mu (N_{\epsilon r}(l))\le \lambda \sum _{l}\mu (N_{r}(l)). \end{aligned}$$

Each $N_r(l)$ is covered by $\Delta _{0}$ and one of its translates $\gamma \Delta _{0}$. By Lemma 5.18, there exists $C'>0$ such that

$$\begin{aligned} \lambda \sum _{l}\mu (N_{r}(l))\le \lambda C'2k \mu (N_r(\Delta _0)). \end{aligned}$$

We can replace $\epsilon$ by $\epsilon ^n$ and using Lemma 5.31 repeatedly, which will yield an arbitrary small $\lambda$ in Lemma 5.29. $\square$

Proof of Lemma 5.31

The proof is similar to the argument of using Lemma 3.11 to deduce Lemma 3.10 in [17]. Let L be the hyperplane containing l and $N_r(L)$ be the r-neighborhood of L. [17, Lemma 3.11] is stated in spherical metric but locally spherical metric is equivalent to the euclidean metric. So [17, Lemma 3.11] implies that there exists $\epsilon >0$ such that for every $\xi \in E:= \Lambda _\Gamma \cap N_{\epsilon r}(l)$, there exists $0<\rho _{\xi }<1$ satisfying

$$\begin{aligned} \mu (B(\xi ,\rho _{\xi })\cap (N_r(L)-N_{\epsilon r}(L)))\ge c\mu (B(\xi ,\rho _{\xi })), \end{aligned}$$

(5.32)

where $0<c<1$ is a constant only depending on $\Gamma$. The family $\{B(\xi ,\rho _{\xi })\}_{\xi \in E}$ forms a covering of E.

It follows from Vitali covering Lemma that there exists a disjoint subcollection $\{B(\xi ,\rho _{\xi }) \}_{\xi \in I}$ with $I\subset E$ countable, such that

$$\begin{aligned} \cup _{\xi \in I} B(\xi ,5\rho _{\xi })\supset \cup _{\xi \in E} B(\xi ,\rho _{\xi })\supset E. \end{aligned}$$

The set $B(\xi ,\rho _{\xi })\cap (N_r(L)-N_{\epsilon r}(L))$ may not be contained in $N_r(l)-N_{\epsilon r}(l)$, but we can cover it by some translations of $N_r(l)-N_{\epsilon r}(l)$. By elementary computation, we can use no more than $k_0$ number of elements $\gamma _j$’s in $\Gamma _{\infty }$ with $k_0$ depending on $\Delta _0$ such that

$$\begin{aligned} \cup _{j}\gamma _j(N_r(l)-N_{\epsilon r}(l))\supset B(\xi ,\rho _\xi )\cap (N_r(L)-N_{\epsilon r}(L)). \end{aligned}$$

Using this inclusion, Lemma 5.18 and disjointness of $B(\xi ,\rho _{\xi })$’s for $\xi \in I$, we obtain

$$\begin{aligned}&C'k_0\mu (N_r(l)-N_{\epsilon r}(l))\ge \mu (\cup _{j}\gamma _j(N_r(l)-N_{\epsilon r}(l)))\\&\quad \ge \sum _{\xi \in I}\mu (B(\xi ,\rho _\xi )\cap (N_r(L)-N_{\epsilon r}(L)) \end{aligned}$$

Using (5.32) and doubling property in Proposition 5.14, we have

$$\begin{aligned}&\sum _{\xi \in I}\mu (B(\xi ,\rho _\xi )\cap (N_r(L)-N_{\epsilon r}(L)) \ge c\sum _{\xi \in I}\mu (B(\xi ,\rho _{\xi }))\\&\quad \ge c\epsilon '\sum _{\xi \in I} \mu (B(\xi ,5\rho _{\xi }))\ge c\epsilon '\mu (N_{\epsilon r}(l)), \end{aligned}$$

Combining the above two formulas, we conclude that there exists $0<\lambda <1$ such that

$$\begin{aligned} \mu (N_{\epsilon r}(l))\le \lambda \mu (N_r(l)). \end{aligned}$$

$\square$

6 Coding of limit set

In this section, we construct the coding and prove Proposition 4.1, Lemma 4.5, and Lemma 4.8. At first reading, the reader might want to concentrate on the case when there is one cusp in the manifold, i.e., ${\mathbf {P}}=\{p_1\}$ and the coordinate change of transformation $g_1$ is the identity. This will significantly reduce the notational burden while not sacrificing too much of the main results.

6.1 Coding for local regions

We introduce “flower” $J_p$, the building block for the coding. Actually, $J_p$’s are almost the union of a countable subcollection of open subsets $\Delta _j$ in the coding. The advantage of considering $J_p$ is that $J_p$ has a clean boundary which makes it possible to estimate the measure.

We first consider the case when $p=\gamma \infty$ is a parabolic fixed point in $\Delta$ with $\gamma \in \Gamma$ the representation of p and $x_{p}=\gamma ^{-1}\infty$. Let $\eta \in (0,1)$. We define the set $J_{p,\eta }$ as follows. By Lemma 5.7, we have

$$\begin{aligned} \gamma ^{-1}B(p,\eta h(p))=B(x_p,1/\eta )^c. \end{aligned}$$

Suppose that $\infty$ is a parabolic fixed point of maximal rank. Then ${\mathbb {R}}^d\subset \partial {\mathbb {H}}^{d+1}$ is tessellated by the translations of ${{\overline{\Delta }}}_0$. Take $R_{p,\eta }$ to be the smallest parallelotope tiled by the translations of ${{\overline{\Delta }}}_0$ such that it contains $B(x_p,1/\eta )$. Let

$$\begin{aligned} J_{p,\eta }= & {} \gamma R_{p,\eta }^c, \end{aligned}$$

(6.1)

$$\begin{aligned} N_p= & {} \{\gamma _1\in \Gamma _{\infty }:\, \gamma _1\Delta _0\subset R_{p,\eta }^c\} =\{\gamma _1\in \Gamma _{\infty }:\,\gamma \gamma _1\Delta _0\subset J_{p,\eta }\}.\nonumber \\ \end{aligned}$$

(6.2)

Suppose $\infty$ is a parabolic fixed point of rank k in general. Let Z be the affine subspace in $\partial {\mathbb {H}}^{d+1}$ described in Lemma 2.3 where elements in $\Gamma _{\infty }$ act as translations, and $\Delta _{0}=B_{Y}(C)\times \Delta _0'$. So Z is tessellated by the translations of $\overline{\Delta _0'}$. Take $R_{p,\eta }$ in Z to be the smallest parallelotope tiled by the translations of $\overline{\Delta _0'}$ such that $B_Y(2/\eta )\times R_{p,\eta }$ contains $B(x_p,1/\eta )$. Set

$$\begin{aligned} J_{p,\eta }= & {} \gamma (B_Y(2/\eta )\times R_{p,\eta })^c\subset B(p,\eta h(p)), \end{aligned}$$

(6.3)

$$\begin{aligned} N_p= & {} \{\gamma _1\in \Gamma _{\infty }:\, \gamma _1\Delta _0\subset (B_Y(2/\eta )\times R_{p,\eta })^c \} \nonumber \\= & {} \{\gamma _1\in \Gamma _{\infty }:\,\gamma \gamma _1\Delta _0\subset J_{p,\eta }\}. \end{aligned}$$

(6.4)

The set $J_{p,\eta }$ enjoys the following property

$$\begin{aligned} J_{p,\eta }\cap \Lambda _\Gamma =\gamma \left( \underset{\gamma _1\in N_p}{\cup }\gamma _1\left( \overline{\Delta _0}\cap \Lambda _{\Gamma }\right) \right) , \end{aligned}$$

(6.5)

that is to say, the countable disjoint union $\underset{\gamma _1\in N_p}{\sqcup }\gamma \gamma _1\Delta _0$ is a conull set in $J_{p,\eta }$. The open sets $\gamma \gamma _1 \Delta _0$ with $\gamma _1\in N_p$ are the ones described in Proposition 4.1 in $J_{p,\eta }$, and on each $\gamma \gamma _1\Delta _0$, the expanding map T is given by $T|_{\gamma \gamma _1\Delta _0}=(\gamma \gamma _1)^{-1}$ (Fig. 2).

We also have the following distance relation:

$$\begin{aligned} d((\gamma \gamma _1)^{-1}\infty ,\Delta _0)=d(\gamma _1^{-1}x_p,\Delta _0)\ge 1/\eta \,\,\,\text {for any}\,\,\, \gamma _1\in N_p. \end{aligned}$$

(6.6)

Fig. 2

The shaded region on the left hand side is $J_{p,\eta }$, which is the image under the action of $\gamma$ on the complement of the white rectangle on the right hand side

Lemma 6.7

There exists $0<c_{4}<1$ such that for any $\eta \in (0,1)$

$$\begin{aligned} B(p,c_{4}\eta h(p))\subset J_{p,\eta }\subset B(p,\eta h(p)), \\B(x_p,1/\eta )\subset (\gamma ^{-1} J_{p,\eta })^c\subset B(x_p, 1/(c_{4}\eta )). \end{aligned}$$

Proof

Due to the compactness of $\Delta _0$, there exists $c_{4}$ such that $(\gamma ^{-1}J_{p,\eta })^c=(B_Y(2/\eta )\times R_{p,\eta })\subset B(x_p,1/(c_{4}\eta ))$. The first statement can be deduce from the second using Lemma 5.7. $\square$

In the following, we abbreviate $J_{p,\eta }$ to $J_p$. For $r>0$, let

$$\begin{aligned}&N_r(\partial J_p):=\{x\in J_p^c:\, d(x,\partial J_p)\le r \},\nonumber \\&N_r(\partial \gamma ^{-1} J_p):=\{x\in (\gamma ^{-1}J_p)^c:\, d(x,\partial \gamma ^{-1}J_p)\le r \}. \end{aligned}$$

(6.8)

Lemma 6.9

Fix $C>1$. For every $0<\eta <1/4C^2$, there exists $0<c=c(\eta )<1$ depending on $\eta$ such that for any $r<h(p)$,

$$\begin{aligned} \mu (N_{C\eta r}(\partial J_p))\le c\mu (N_r(\partial J_p)). \end{aligned}$$

(6.10)

Moreover, $c(\eta )$ tends to zero as $\eta$ tends to zero.

The proof of Lemma 6.9 will be given in the “Appendix”.

We consider the general case. Let p be any parabolic fixed point in $\Delta$. Write $p=\gamma p_i$ with $\gamma \in \Gamma$ the representation of p. If $p_i=\infty$, let $J_p$ and $N_p$ be defined as (6.3) and (6.4) respectively. Otherwise, we use the following commutative diagram to define $J_p$:

Note that $g_ip=(g_i\gamma g_i^{-1})\infty \in g_i\Delta$. So for the action of $g_i\Gamma g_i^{-1}$ on $\partial {\mathbb {H}}^{d+1}$, we can define $J_{i,p}$ for the parabolic fixed point $g_ip$ as (6.3). Set

$$\begin{aligned} J_p:=g_i^{-1}J_{i,p},\ \ N_p=\{\gamma _1\in \Gamma _{p_i}:\,\gamma \gamma _1\Delta _{p_i}\subset J_{p}\}, \end{aligned}$$

(6.11)

where $\Gamma _{p_i}$ is a subgroup of $\Gamma$ defined in (5.9). The set $J_p$ enjoys the property

$$\begin{aligned} J_{p}\cap \Lambda _\Gamma =\underset{\gamma _1\in N_p}{\sqcup }\left( \gamma \gamma _1{\overline{\Delta }}_{p_i}\cap \Lambda _\Gamma \right) . \end{aligned}$$

(6.12)

On each set $\gamma \gamma _1\Delta _{p_i}$, we have an expanding map given by $(\gamma \gamma _1)^{-1}$ which maps this set to $\Delta _{p_i}$.

The following lemma is an analog of Lemma 6.7.

Lemma 6.13

There exists some constant $C_{5}>1$ such that for any $\eta \in (0,1)$

$$\begin{aligned} B(g_ix_p,1/\eta )&\subset (g_i\gamma ^{-1}J_p)^c\subset B(g_ix_p,1/(c_{4}\eta )),\\ B\left( p,\eta h(p)/C_{5}\right)&\subset g_i^{-1}B\left( g_ip, c_{4}\eta h(g_i p)\right) \subset J_p\subset g_i^{-1}B\left( g_ip,\eta h(g_i p)\right) \\&\subset B\left( p, C_{5}\eta h(p)\right) , \end{aligned}$$

where $x_p=\gamma ^{-1}p_i$.

Proof

We use Lemma 6.7, 5.11 and 5.12 to obtain the lemma. $\square$

6.2 Coding for $\Delta _0$

The construction of the coding for the whole $\Delta _0$ is by induction. Let $\Omega _0:=\Delta _{0}$.

Since we already have a nice coding for flowers $J_p$, the idea is to find a collection of pairwise disjoint flowers $J_p$ to cover the intersection $\Lambda _{\Gamma }\cap \Omega _0$. Here p is a parabolic fixed point. From the construction of $J_p$ (Lemma 6.7), we know that the higher the height h(p) is, the larger $J_p$ is. So we start with parabolic fixed points with large heights. We want that the full flower $J_p$ is inside $\Omega _0$. Hence we only take parabolic fixed points p away from the boundary of $\Omega _0$.

Take

$$\begin{aligned}&h_n=e^{-n},\\&\eta \in (0,1)\text { a sufficiently small constant to be specified at the end of the proof}\\&\quad \text { of Proposition} 6.15. \end{aligned}$$

All the constants appearing later will be independent of $\eta$ unless we state it explicitly.

• For $n\in {\mathbb {N}}$, let

  $$\begin{aligned} P_{n+1}=\{p\in {{\mathcal {P}}}:\,\ \eta h(p)\in (h_{n+1},h_n],\ B(p,h_n/(4\eta ))\subset \Omega _{n} \}. \end{aligned}$$

  (6.14)

• For any $p\in P_{n+1}$, write $p=\gamma p_i$ with $\gamma \in \Gamma$ the representation of p. Construct $J_p$ and $N_p$ as in the previous section.

• Set

  $$\begin{aligned}&\Omega _{n+1}=\Omega _n-D_{n+1}=\Omega _n-\cup _{p\in P_{n+1}}J_{p}. \end{aligned}$$

Using the definition of $J_p$, Lemma 6.13 and the separation property (Lemma 6.16), it can be shown that the sets $J_p$’s with $p\in P_n$ and $n\in {\mathbb {N}}$ are mutually disjoint (Lemma 6.17) and inside $\Delta _{0}$. In Proposition 6.15, it will be shown that the union $\cup _n\cup _{p\in P_n}J_p$ is conull in $\Delta _0$ with respect to the PS measure $\mu$. By (6.12), the countable disjoint union

$$\begin{aligned} \bigcup _{n\in {\mathbb {N}}}\bigcup _{p=\gamma p_i\in P_n}\bigcup _{\gamma _1\in N_p}\gamma \gamma _1\Delta _{p_i} \end{aligned}$$

is also conull in $\Delta _0$ with respect to PS measure. On each set $\gamma \gamma _1\Delta _{p_i}$ we have an expanding map given by $(\gamma \gamma _1)^{-1}$ which maps this set to $\Delta _{p_i}$. For one cusp case, these are the countable collection of disjoint open subsets and the expanding map. When there are multi-cusps, this is the first step to construct the coding and the rest will be provided in Sects. 6.7 and 6.8.

The main result of this section is the following proposition.

Proposition 6.15

There exist $\epsilon _0>0$ and $N>0$ such that for all $n>N$, we have

$$\begin{aligned} \mu (\Omega _n)\le (1-\epsilon _0)^n. \end{aligned}$$

For one cusp case, this yields Proposition 4.1 (1). Moreover, the exponential tail property (4.2) will follow from Proposition 6.15 rather directly and it will be proved in Sect. 6.7. To prove this proposition, we need a lot of preparations and we postpone its proof to the end of Sect. 6.6.

6.3 Separation

Lemma 6.16

(Separation property) For any two different parabolic fixed points $p, p'$, we have

$$\begin{aligned} d(p,p')>\sqrt{h(p) h(p')}. \end{aligned}$$

Proof

Let $x,x'$ be the euclidean centers of $H_p$ and $H_{p'}$ repectively. By disjointness of horoballs, then $d_E(x,x')\ge (h(p)+h(p')/2$. By the Pythagoras’ theorem, we obtain

$$\begin{aligned} d(p,p')\ge \sqrt{d_E(x,x')^2-((h(p)-h(p'))/2)^2}\ge \sqrt{h(p)h(p')}. \end{aligned}$$

$\square$

This property plays a key role in the construction of the coding and the proof of Proposition 6.15.

Lemma 6.17

If $\eta <1/(4eC_{5})$, then the sets $J_p$’s with $p\in P_n$ and $n\in {\mathbb {N}}$ are mutually disjoint, and the distance between any two connected components of $\partial \Omega _n$ is strictly greater than $h_n/(2\eta )$.

Proof

Notice that $\Omega _{n}=\Omega _{n-1}-\cup _{p\in P_n} J_p$. By induction, we only need to prove two cases.

Case 1: We consider $J_p,J_{p'}$ with distinct $p,p'$ in $P_n$. Using Lemma 6.13 and Lemma 6.16, we obtain

$$\begin{aligned} \sqrt{h(p)h(p')}-C_{5}\eta (h(p)+h(p'))\ge \frac{h_n}{\eta }-2C_{5} h_{n-1}\ge \frac{h_n}{2\eta }. \end{aligned}$$

Therefore, two sets $J_p, J_{p'}$ are disjoint and the distance between them is at least $h_n/(2\eta )$.

Case 2: We consider $J_p$ with $p\in P_n$ and $\Omega _{n-1}$. By (6.14) and Lemma 6.13, the distance between $J_{p}$ and $\partial \Omega _{n-1}$ is also greater than

$$\begin{aligned} \frac{h_{n-1}}{4\eta } -C_{5}\eta h(p) \ge \frac{h_{n-1}}{4\eta } -C_{5}h_{n-1}>\frac{h_n}{2\eta }. \end{aligned}$$

Hence $J_p$ is inside $\Omega _{n-1}$.

By induction, for different connected components of $\Omega _{n}$, their distance is at least $h_{n}/(2\eta )$. $\square$

6.4 Equivalence classes in $Q_n$

Motivation of equivalence classes. We introduce the notion of equivalence classes to attain the exponential tail property.

Let’s start with some definitions. The visual map $\pi :{\text {T}}^1({\mathbb {H}}^{d+1})\rightarrow \partial {\mathbb {H}}^{d+1}$ is defined by

$$\begin{aligned} \pi (x)=\lim _{t\rightarrow \infty } {\mathcal {G}}_t(x), \end{aligned}$$

which maps x to the forward endpoint in $\partial {\mathbb {H}}^{d+1}$ of the geodesic defined by x.

Recall that we fix $p_1$ as $\infty$ and $H_{\infty }$ is the horoball based at $\infty$ given by ${\mathbb {R}}^d\times \{x\in {\mathbb {R}}:\,x>1\}$. Let $\widetilde{H_{\infty }}$ be the corresponding unstable horosphere. More precisely, let $x_o$ be the unit tangent vector based at $(0,1)\in {\mathbb {R}}^d\times {\mathbb {R}}$ with $\pi (x_o)=0$. Then $\widetilde{H_{\infty }}$ is the set of x in ${\text {T}}^1({\mathbb {H}}^{d+1})$ such that $d({{\mathcal {G}}}_{-t}x_o,{{\mathcal {G}}}_{-t}x)\rightarrow 0$ as $t\rightarrow +\infty$. For a set $E\subset \partial {\mathbb {H}}^{d+1}-\{\infty \}\simeq {\mathbb {R}}^d$, let

$$\begin{aligned} {\widetilde{E}}=\pi |_{\widetilde{H_\infty }}^{-1}(E) \end{aligned}$$

be the preimage of E under the map $\pi$ restricted on $\widetilde{H_\infty }$.

Let

$$\begin{aligned}&H_p(\eta )\,\,\, \text {be the horoball based at}\,\,\, p\,\,\, \text {with height equal to}\,\,\, \eta h(p),\\&{\mathcal {C}}_\eta =\cup _{p\in {{\mathcal {P}}}}\Gamma {\text {T}}^1(H_p(\eta )). \end{aligned}$$

Then ${{\mathcal {C}}}_\eta$ is the lift of the unit tangent bundle over proper horocusps of M.

At the time n, the set ${{\mathcal {G}}}_n\widetilde{\Omega _n}$ is a large sheet with many holes, consisting of “flowers” of different sizes, corresponding to different ${{\mathcal {G}}}_n \widetilde{J_p}$. Here is the source of the exponential tail: for $x\in {{\mathcal {G}}}_n\widetilde{\Omega _n}$ with $\pi (x)\in \Lambda _{\Gamma }\cap \Omega _0$ and x not in the cusp region ${{\mathcal {C}}}_\eta$, the recurrence of geodesic flow implies that there exits a new flower ${{\mathcal {G}}}_n\widetilde{ J_p}$ inside the neighborhood of x with size bounded below (Lemma 6.55). So a fixed proportion of the neighborhood of x will be coded in a fixed time (Lemma 6.53).

Lemma 6.55 doesn’t apply to the set ${{\mathcal {G}}}_n\widetilde{\Omega _n}\cap {{\mathcal {C}}}_\eta$. Points in the set $\pi ({{\mathcal {G}}}_n\widetilde{\Omega _n}\cap {{\mathcal {C}}}_\eta )$ are contained in the balls centered at certain parabolic fixed points. We want to argue that from step n to step $n+1$, the points near the outer edge of the balls will escape the cusps. We illustrate the scheme of the proof. For $n\in {\mathbb {N}}$, define

$$\begin{aligned} Q_{n+1}= & {} \{p\in {{\mathcal {P}}}:\, \eta h(p)\in (h_{n+1},h_n],\ B(p,\eta h(p))\cap \Omega _n\ne \emptyset ,\\&\quad d(p,\partial \Omega _n)\le h_n/(4\eta )\}. \end{aligned}$$

Then we can cover ${{\mathcal {C}}}_\eta \cap {{\mathcal {G}}}_n\widetilde{\Omega _n}$ by the union of balls centered at parabolic fixed points in $Q_m$ with $m\le n$ (Lemma 6.37, 6.38):

$$\begin{aligned} {{\mathcal {C}}}_\eta \cap {{\mathcal {G}}}_n\widetilde{\Omega _n}={{\mathcal {G}}}_n(\cup _{m\le n}\cup _{p\in Q_m}{\widetilde{B}}(p,r(p,n)))\cap {{\mathcal {G}}}_n\widetilde{\Omega _n}, \end{aligned}$$

where $r(p,n)=\sqrt{\eta h(p)h_n}$. From step n to step $n+1$, the part

$$\begin{aligned} {{\mathcal {G}}}_{n+1}({\widetilde{B}}(p,r(p,n))-{\widetilde{B}}(p,r(p,n+1))) \end{aligned}$$

will leave the cusp region ${{\mathcal {C}}}_\eta$. We want a lower bound for the measure of $({B}(p,r(p,n))-{B}(p,r(p,n+1)))\cap \Omega _{n}$. The main difficulty is that $B(p,r(p,n))\cap \Omega _{n}$ may not be a full ball, in which case its PS measure is hard to estimate.

We introduce the notion of equivalence classes to resolve this issue. Consider a subset of $Q_n$:

$$\begin{aligned} Q_{n}'=\{p\in Q_{n}:\,\ B(p,r(p,n))\cap \partial \Omega _{n-1}\ne \emptyset \}. \end{aligned}$$

Pick any $p\in Q_n'$. As the ball $B(p,r(p,n))\cap \Omega _{n-1}$ is not a full ball, we will pair it with another partial ball and use the doubling property of the PS measure. Notice that there is a unique component in $\partial \Omega _{n-1}$ closest to p (Lemma 6.17). If it is $\partial \Omega _0$, note that $\Lambda _{\Gamma }\cap {\mathbb {R}}^d$ is covered by the translations of $\Delta _0$, so the symmetry property of these translations gives the point $p'$ to pair with p (if p is around the corners of $\partial \Omega _0$, we may need more than one point to pair with p). If it is some $\partial J_q$, write $q=\gamma ^{-1}p_i$ with $\gamma ^{-1}\in \Gamma$ the representation of q. We map B(p, r(p, n)) and $\partial J_q$ by $g_i\gamma$ and get a picture similar to the previous case. We find the paring point for $g_i\gamma p$ and map it back to get the one for p (Fig. 3). The work lies in modifying the radius r(p, n): r(p, n) is defined to $\sqrt{\eta h_n h(p)}$ depending on h(p), and it may happen that the horosphere attached to the pairing point of p has a different height.

Fig. 3

Pairing partial balls: $q=\gamma ^{-1}p_i$, $p'=g_i\gamma p$, $B_1=B(g_i\gamma p, {{\tilde{r}}}_{p,m}/C_{6})$, $\gamma _1\in (g_i\Gamma g_i^{-1})_{\infty }$

Finding the radius. For Lemma 6.18 - Lemma 6.23, we consider $p\in Q_n'$ such that the component in $\partial \Omega _{n-1}$ closest to p is $\partial J_{q}$ with $q\in \cup _{l=1}^{n-1}P_l$. Write $q= \gamma ^{-1}p_i$ for some $\gamma \in \Gamma$ the representation of q (Fig. 3).

Lemma 6.18

There exists $C>1$ such that we have

$$\begin{aligned} \eta h(p)\le h_{n-1}\le C\eta ^3 h(q),\ \ \frac{\eta h(g_i p)}{C}\le h_{n-1}\le C \eta ^3 h(g_i q). \end{aligned}$$

Proof

It follows from Lemma 6.16 that

$$\begin{aligned} d(p,q)\ge \sqrt{h(p) h(q)}\ge \sqrt{h_{n-1} h(q)/(e\eta )}. \end{aligned}$$

Meanwhile by Lemma 6.13, we have

$$\begin{aligned} d(p,q)\le & {} d(p,\partial J_{q})+\max _{y\in \partial J_q}d(y, q)\le r(n,p)\\&+C_5\eta h(q)\le h_{n-1}+C_{5}\eta h(q)\le (1+C_{5})\eta h(q). \end{aligned}$$

So the above two inequalities lead to first statement. The second statement follows easily from the first statement and Lemma 5.11. $\square$

Lemma 6.19

There exists $C>1$ such that

$$\begin{aligned} B(g_ip, h(g_i p))\subset B(g_iq, C\eta h(g_i q))-B(g_iq, \eta h(g_i q)/C). \end{aligned}$$

Proof

For any $\xi \in \partial B(g_ip, h(g_i p))$, an upper bound for $d(\xi , g_iq)$ is given by

$$\begin{aligned} d(\xi , g_iq)&\le d(\xi , g_ip)+d(g_ip,\partial J_{g_iq})+\max _{y\in \partial J_{g_iq} }d(y,g_iq)\\&\le h(g_i p)+C h_{n-1}+\eta h(g_i q) \le C\eta h(g_i q). \end{aligned}$$

A lower bound for $d(\xi , g_iq)$ is given by

$$\begin{aligned} d(\xi , g_iq)&\ge d(g_iq, \partial J_{g_iq}) -d(g_ip,\partial J_{g_iq})-h(g_i p)\\&\ge c_{4} \eta h(g_i q) -Ch_{n-1}-h(g_i p) \ge c_{4} \eta h(g_i q) -C\eta ^2h(g_i q). \end{aligned}$$

Hence by taking $\eta$ sufficiently small, we reach the conclusion. $\square$

Lemma 6.20

There exists $C>1$ such that we have

$$\begin{aligned} \frac{ h(g_i p)}{C\eta ^2 h(g_i q)}\le h(g_i \gamma p)\le \frac{Ch(g_i p)}{\eta ^2 h(g_i q)}. \end{aligned}$$

Proof

Apply Lemma 5.4 to the horoball $H=H_{g_ip}$ based at $g_ip$ and the element $g=g_i\gamma g_i^{-1}$. Notice that $g_iq=(g_i\gamma g_i^{-1})^{-1}\infty =g^{-1}\infty$ and $h(g_i q)=h(H_{g_iq})=h(g^{-1}H_\infty )$. We have

$$\begin{aligned} \frac{h(g_i q)h(g_i p)}{d(g_iq,g_ip)^2+h(g_i p)^2}\le h(g_i \gamma p)=h(gH)\le \frac{h(g_i q)h(g_i p)}{(d(g_iq,g_ip)-h(g_i p) /2)^2}, \end{aligned}$$

By Lemma 6.18 and 6.19, we obtain the lemma. $\square$

For any $m\in {\mathbb {N}}$, set

$$\begin{aligned} {\tilde{r}}_{p,m}=\sqrt{\frac{h_mh(g_i \gamma p)}{\eta h(g_i q)}}. \end{aligned}$$

(6.21)

At the point $g_i\gamma p$, we will consider ball $B(g_i\gamma p, {{\tilde{r}}}_{p,m}/C_{6})$, where $C_{6}>1$ is a constant given in Lemma 6.23 such that ${\tilde{r}}_{p,m}/C_{6}$ guarantees the equivalence classes we introduce later are well-defined. Once we have chosen the ball $B(g_i\gamma p, {{\tilde{r}}}_{p,m}/C_{6})$, we will map it back by $(g_i\gamma )^{-1}$ to attain the “correct” ball at p.

By Lemma 6.20, 5.11 and 6.18, we have

$$\begin{aligned} {\tilde{r}}_{p,n}\le \frac{C\sqrt{\eta h_nh(g_i p)}}{\eta ^2h(g_i q)}\le \frac{Ch(g_i p)}{\eta h(g_i q)}\le C\eta h(g_i\gamma p). \end{aligned}$$

(6.22)

Lemma 6.23

There exists $C_{6}>1$ such that for any point $p'$, if $d(g_i\gamma p',g_i\gamma \partial J_q)\le {\tilde{r}}_{p,n}/C_{6}$, then $d(p',\partial J_q)\le h_n$.

Proof

It follows from Lemma 6.7 applying $g_i\gamma g_i^{-1}$ that for any $C_{6}>1$, if $d(g_i\gamma p', g_i\gamma \partial J_q)\le {\tilde{r}}_{p,n}/C_{6}$, then

$$\begin{aligned} g_i\gamma p'=g_i\gamma g_i^{-1}(g_ip')\in B(g_ix_q, 2/(c_{4}\eta ))-B(g_ix_q,1/(2\eta )), \end{aligned}$$

where $g_ix_q=g_i\gamma p_i=g_i\gamma g_i^{-1}\infty$. For any x in the line segment between $g_i\gamma p'$ and $g_i\gamma \partial J_{q}$, we use Lemma 5.7 to obtain $|(g_i\gamma ^{-1} g_{i}^{-1})'x|\le 4\eta ^2 h(g_i q)$.

By Lemma 5.12 and (6.22), we obtain

$$\begin{aligned} d(p',\partial J_q)&\le Cd(g_ip',g_i\partial J_{q})\le C\eta ^2 h(g_i q) d(g_i\gamma p',g_i\gamma \partial J_q)\\&\le C\eta ^2{\tilde{r}}_{p,n}h(g_i q)/C_{6} \le C\eta h(g_i p)/C_{6}. \end{aligned}$$

By taking $C_{6}>1$ large enough, we have $d(p', \partial J_q)\le h_n$. $\square$

Definition of equivalence classes. Now we define equivalence classes in $Q_n$. We define them by induction. For $Q_1$,

• for $p\in Q_1-Q_1'$, set the equivalence class C(p) of p to be $\{p\}$.

• for $p\in Q_1'$, set

  $$\begin{aligned} C(p)=\{\gamma _1 p:\,\gamma _1 B(p,\sqrt{\eta h_1 h(p)})\cap \partial \Omega _0\ne \emptyset ,\,\,\, \gamma _1\in \Gamma _{\infty }\}. \end{aligned}$$

Set

$$\begin{aligned} Q_1'':=\cup _{p\in Q_1} C(p), \end{aligned}$$

and for any $p'\in Q_1''$ and $m\ge 1$, define

$$\begin{aligned} r_{p',m}=\sqrt{\eta h_m h(p')},\ \ B_{p',m}=B(p,r_{p',m}). \end{aligned}$$

Suppose we have defined $Q_n''$. We define the equivalence classes in $Q_{n+1}$ and the set $Q_{n+1}''$ as follows:

Case 1::

for $p\in Q_{n+1}'-\cup _{l\le n}Q_l''$ such that the component in $\partial \Omega _n$ closest to p is $\partial \Omega _0$, set

$$\begin{aligned} C(p)=\{\gamma _1p:\, \gamma _1B(p,\sqrt{\eta h_{n+1}h(p)})\cap \partial \Omega _0\ne \emptyset ,\,\, \gamma _1\in \Gamma _{\infty }\}. \end{aligned}$$

For any $p'\in C(p)$ and $m\ge n+1$, define

$$\begin{aligned} r_{p',m}=\sqrt{\eta h_m h(p')},\ \ B_{p',m}=B(p',r_{p',m}). \end{aligned}$$

Case 2::

for $p\in Q_{n+1}'-\cup _{l\le n}Q_l''$ such that the component in $\partial \Omega _n$ closest to p is some $J_{q}$, write $q=\gamma ^{-1}p_i$ with $\gamma ^{-1}$ the representation of q. Let ${\tilde{r}}_{p,n}$ and $C_{6}$ be as given in (6.21) and Lemma 6.23 respectively. If $B(g_i\gamma p,{\tilde{r}}_{p,n}/C_{6})\cap \,g_i\gamma \partial J_q\ne \emptyset$, set

$$\begin{aligned}&C(p)= \{(g_i\gamma )^{-1} \gamma _1g_i\gamma p:\, \gamma _1B(g_i\gamma p,{\tilde{r}}_{p,n}/C_{6})\cap \,g_i\gamma \partial J_q\ne \emptyset , \nonumber \\&\quad \qquad \gamma _1\in (g_i\Gamma g_i^{-1})_{\infty }\}. \end{aligned}$$

(6.24)

Otherwise, set $C(p)=\{p\}$. For any $p'\in C(p)$ and $m\ge n+1$, define

$$\begin{aligned}&r_{p',m}=\frac{1}{C_{6}}\sqrt{\frac{h_m h(g_i \gamma p')}{\eta h(g_i q)}}\,\,\,(\text {which equals}\,\,\,r_{p,m}), \end{aligned}$$

(6.25)

$$\begin{aligned}&B_{p',m}=(g_i\gamma )^{-1}B(g_i\gamma p', r_{p',m}). \end{aligned}$$

(6.26)

Case 3::

for $p\in Q_{n+1}- \cup _{l\le n}Q_l''$ such that p does not belong to the union of equivalence classes defined in the previous two cases, set $C(p)=\{p\}$ and for any $m\ge n+1$, define

$$\begin{aligned} r_{p,m}=\sqrt{\eta h_m h(p)},\ \ B_{p,m}=B(p,r_{p,m}). \end{aligned}$$

Set

$$\begin{aligned} Q_{n+1}''=\bigcup _{p\in Q_{n+1}-\cup _{l\le n}Q_l''}C(p). \end{aligned}$$

Then $\cup _{1\le l\le (n+1)}Q_l''\supset \cup _{1\le l\le (n+1)}Q_l$.

It is worthwhile to point out that under our definition of equivalence classes, it may happen that for $p\in Q_n-\cup _{l<n}Q_l''$, its equivalence class C(p) may contain points $p'$ whose associated horospheres don’t appear in the time interval $[n-1,n)$. This is our motivation to establish results like Lemmas 6.27 and 6.40.

In the following discussion of the points p in $Q_{n}''$, if the definition of p involves a boundary component of $\partial \Omega _{n-1}$, we will need to consider this boundary component a lot of the times. For simplicity, we call the boundary component used to define $p\in Q_{n}''$ the associated boundary component of p .

Uniformity among equivalence classes. For $p\in Q_{n}-\cup _{l<n} Q_l''$, we show that, up to some constant, the points in the equivalence class C(p) are “uniform".

Lemma 6.27

There exists $C_{7}>1$ such that for any $p\in Q_{n}-\cup _{l<n} Q_l''$ and any $p'\in C(p)$ we have

$$\begin{aligned} 1/C_{7}\le h(p)/h(p')\le C_{7}. \end{aligned}$$

It suffices to prove Lemma 6.27 for the case when $\# C(p)\ge 2$ and the associated component of p in $\partial \Omega _{n-1}$ is some $\partial J_q$. Write $q=\gamma ^{-1}p_i$ with $\gamma ^{-1}$ the representation of q. Let $r_{p,m}$ and $B_{p,m}$ be defined as in (6.25) and (6.26) respectively. We first show the following estimate.

Lemma 6.28

(Location of balls) There exists a constant $C>1$ such that for C(p) with the associated boundary component $\partial J_q$ and for $p'\in C(p)$

$$\begin{aligned}&B(g_i\gamma p', r_{p',m})\subset B \left( g_ix_q, C/\eta \right) -B(g_ix_q,1/(C\eta )), \end{aligned}$$

(6.29)

$$\begin{aligned}&g_iB_{p',m}\subset B(g_iq, C\eta h(g_i q))-B(g_iq,\eta h(g_i q)/C). \end{aligned}$$

(6.30)

Proof

By Lemma 6.13, we have

$$\begin{aligned} g_i\gamma \partial J_q\subset B(g_i x_q, 1/(c_4\eta ))-B(g_i x_q, 1/\eta ). \end{aligned}$$

We also have $r_{p,n}\le C\eta$ by (6.22). Meanwhile, the construction of the equivalence class C(p) implies that $r_{p',m}=r_{p,m}$. Hence we obtain (6.29). We use Lemma 5.7 to obtain (6.30) from (6.29). $\square$

Proof of Lemma 6.27

We prove the following explicit estimate:

$$\begin{aligned} h(g_i p')\approx \eta ^2 h(g_i q) h(g_i \gamma p'). \end{aligned}$$

(6.31)

This together with Lemma 5.11 and $h(g_i \gamma p')=h(g_i \gamma p)$ will lead to Lemma 6.27. Note that $h(g_i \gamma p')\le C$, with C a constant depending on $\Gamma$. We apply Lemma 5.4 to the horoball $H=H_{g_i\gamma p'}$ based at $g_i\gamma p'$ and the element $g=g_i\gamma ^{-1}g_i^{-1}$. Notice that $g_ix_q =(g_i\gamma ^{-1}g_i^{-1})^{-1}\infty =g^{-1}\infty$ and $h(g_i x_q)=h(g^{-1} H_\infty )$. We obtain

$$\begin{aligned}&\frac{h(g_i x_q) h(g_i \gamma p')}{d(g_ix_q,g_i\gamma p')^2+h(g_i \gamma p')^2} \le h(g_i p')\\&\quad =h(gH) \le \frac{h(g_i x_q) h(g_i \gamma p')}{(d(g_ix_q, g_i\gamma p')-h(g_i \gamma p')/2)^2}. \end{aligned}$$

Due to (5.2), we have $h(g_ix_q)=h(g^{-1}H_\infty )=h(gH_\infty )=h(g_iq)$. By (6.29), we have $d(g_i\gamma p',g_ix_q)\approx 1/\eta$. Therefore, we obtain

$$\begin{aligned} \frac{\eta ^2 h(g_i q) h(g_i \gamma p')}{C}\le h(g_i p')\le C\eta ^2 h(g_i q) h(g_i \gamma p'). \end{aligned}$$

$\square$

Lemma 6.32

There exists $C_{8}>1$ such that for any $p\in Q''_{n}$ and any $m\ge n$, the ball $B_{p,m}$ satisfies

$$\begin{aligned} B(p, \sqrt{\eta h(p) h_m}/C_{8})\subset B_{p,m} \subset B(p, C_{8}\sqrt{\eta h(p) h_m}). \end{aligned}$$

Proof

It is enough to prove the case when the associated component of p in $\partial \Omega _{n-1}$ is some $\partial J_q$. Write $q=\gamma ^{-1}p_i$ with $\gamma ^{-1}$ the representation of q. By definition,

$$\begin{aligned} B_{p,m}=(g_i\gamma )^{-1}B(g_i\gamma p, r_{p,m}) \end{aligned}$$

with $r_{p,m}=\frac{1}{C_{6}}\sqrt{\frac{h_m h(g_i\gamma p)}{\eta h(g_i q)}}$.

Consider the action of $g_i\gamma ^{-1} g_i^{-1}$ on $B(g_i\gamma p, r_{p,m})$. By Lemma 5.7 and (6.29), we have

$$\begin{aligned} |(g_i\gamma ^{-1}g_i^{-1})'x|\approx \eta ^2 h(g_iq)\,\,\,\text {for any}\,\,x\in B(g_i\gamma p, r_{p,m}). \end{aligned}$$

(6.33)

Meanwhile, there exists a point $p'\in C(p)$ such that $p'$ satisfies Lemma 6.20. We have $h(g_i\gamma p)=h(g_i\gamma p')$ and $h(p)\approx h(p')$ (Lemma 6.27). As a result, we obtain

$$\begin{aligned} h(g_i\gamma p)\approx \frac{h(g_i p)}{\eta ^2 h(g_i q)}. \end{aligned}$$

(6.34)

(6.33) and (6.34) yield there exists $C>1$ such that

$$\begin{aligned} B(g_ip, \sqrt{\eta h(g_ip) h_m}/C)\subset g_iB_{p,m} \subset B(g_ip, C\sqrt{\eta h(g_ip) h_m}). \end{aligned}$$

We use Lemma 5.11 and 5.12 to finish the proof. $\square$

Well-definedness of equivalence classes

Lemma 6.35

For any two equivalence classes $C(p')$ and $C(p'')$, they are either the same or disjoint.

Proof

Assume that these two equivalence classes are not the same and the intersection is nonempty.

Case 1: Suppose one of these two equivalence classes just consists of one point, say $\#C(p')=1$ and $\#C(p'')\ge 2$. We may assume that $p''\in Q_n'$ for some n. If the associated component of $p''$ is $\partial \Omega _0$, then we are in Case 1 of the definition of equivalence classes. For $p'\in C(p'')$, we obtain that $h(p')=h(p'')\in (h_n,h_{n-1}]/\eta$. As $\#C(p')=1$, we know that $p'$ is contained in $Q_l-\cup _{i<l}Q_i''$ for some $l<n$, which contradicts $h(p')\in (h_n,h_{n-1}]/\eta$.

In the sequel, we assume further that the associated component of $p''$ is some $\partial J_q$. Write $q=\gamma ^{-1}p_i$. As $p'$ belongs to the equivalence class $C(p'')$, it follows from the definition that

$$\begin{aligned} B(g_i\gamma p', r_{p',n})\cap g_i \gamma \partial J_q\ne \emptyset ,\,\,\, B(g_i\gamma p'', r_{p'',n})\cap g_i \gamma \partial J_q\ne \emptyset ,\nonumber \\ \end{aligned}$$

(6.36)

where $r_{p',n}$ is defined as in (6.25) and equals $r_{p'',n}$.

The fact that $C(p')$ just consists of $p'$ implies $p'\in Q_l-\cup _{i<l}Q_i''$ for some $l<n$. Meanwhile, as $\partial J_q$ is the associated component of $p''$, by Lemma 6.18 and 6.27, we have

$$\begin{aligned} h(q)\ge h(p'')/(C\eta ^2)\ge h(p')/(C\eta ^2)\ge h_l/(C\eta ^3). \end{aligned}$$

Hence $\partial J_q\subset \partial \Omega _l$.

(6.36) allows us to apply Lemma 6.23 to $p'$, and we obtain

$$\begin{aligned} d(p',\partial J_q)\le h_n< \sqrt{\eta h_l h(p')}. \end{aligned}$$

So $p'\in Q_l'-\cup _{i<l}Q_i''$. (6.36) yields

$$\begin{aligned} B(g_i\gamma p',r_{p',l})\cap g_i\gamma \partial J_q\ne \emptyset ,\,\,\, B(g_i\gamma p'',r_{p'',l})\cap g_i\gamma \partial J_q\ne \emptyset . \end{aligned}$$

As $l<n$, $C(p')$ contains $p''$, which is a contradiction.

Case 2: Suppose that $\# C(p'),\,\#C(p'')\ge 2$. Without loss of generality, we may assume that $p'\in Q_m'-\cup _{l<m}Q_l''$ and $p''\in Q_n'-\cup _{l<n}Q_l''$ and $m\le n$.

Let $p\in C(p')\cap C(p'')$. Then it follows from the construction of equivalence classes and Lemma 6.23 that there are boundary components $\partial _1$ and $\partial _2$ in $\partial \Omega _{n-1}$ such that

$$\begin{aligned} d(p,\partial _1)\le h_{m-1},\,\,\,\ d(p,\partial _2)\le h_{n-1}. \end{aligned}$$

On the one hand, as $\partial _1$ and $\partial _2$ are in $\partial \Omega _{n-1}$, if they are distinct, Lemma 6.17 states that their distance is greater than $h_{n-1}/(2\eta )$. On the other hand, using Lemma 6.27, we obtain

$$\begin{aligned} h_n/h_m\ge h(p'')/(eh(p'))=(h(p'')/h(p))(h(p)/(eh(p')))\ge 1/(eC_{7}^2). \end{aligned}$$

Then

$$\begin{aligned} d(\partial _1,\partial _2)\le h_{m-1}+h_{n-1}\le (1+eC_{7}^2)h_{n-1}<h_{n-1}/(2\eta ). \end{aligned}$$

We conclude that $\partial _1=\partial _2$.

There are two possibilities for $\partial _1$. One possibility is that $\partial _1$ is some $\partial J_q$. Write $q=\gamma ^{-1}p_i$ with $\gamma ^{-1}$ the representation of q. As $g_i\gamma p$ is related with $g_i\gamma p'$ and $g_i\gamma p''$ by elements in $(g_i\Gamma g_i^{-1})_{\infty }$, we have $\gamma _1 g_i\gamma p''=g_i\gamma p'$ for some $\gamma _1\in (g_i\Gamma g_i^{-1})_{\infty }$. As $m\le n$, we have

$$\begin{aligned} \emptyset \ne B(g_ip'', r_{p'',n})\cap g_i\gamma \partial J_{q}\subset B(g_ip'', r_{p'',m})\cap g_i \gamma \partial J_{q}. \end{aligned}$$

As a result, we have $C(p')= C(p'')$. The other possibility is that $\partial _1=\partial \Omega _0$. It follows directly from the construction of equivalence classes that

$$\begin{aligned} \emptyset \ne B(p'', r_{p'',n})\cap \partial \Omega _0\subset B(p'', r_{p'',m})\cap \partial \Omega _0. \end{aligned}$$

Hence $C(p')=C(p'')$. $\square$

6.5 Auxiliary sets $A_n$ and $B_n$ in $\Omega _n$

We introduce auxiliary sets $A_n$ and $B_n$ in $\Omega _n$. By Lemma 6.35, the set $Q_n''$ is disjoint with $\cup _{1\le l\le (n-1)}Q_n''$. For any $p\in Q_n''$ and any $m\ge n$, we have defined the ball $B_{p,m}$. Note that it follows from the construction of $Q_{n}''$ that if $\#C(p)=1$, then the full ball $B_{p,n}$ is contained in $\Omega _n$. For each n, we define

$$\begin{aligned} B_{n}=\Omega _n\cap \bigcup _{p\in \cup _{1\le l\le n}Q_l''} B_{p,n}\,\,\,\text {and}\,\,\,\ \ A_{n}=\Omega _{n}-B_{n}. \end{aligned}$$

In the followings, we will show how to use the set $B_n$ to detect whether a point is in the cusps of the manifold at time $t=n$ or not.

6.5.1 $B_n$ and cusps

Lemma 6.37

For $x\in \widetilde{\Omega _0}$, if ${\mathcal {G}}_n x\in {\mathcal {C}}_\eta$, then there exists a parabolic fixed point p with $\eta h(p)>h_n$ such that

$$\begin{aligned} d(\pi (x),p)<\sqrt{\eta h(p)h_n}. \end{aligned}$$

Proof

By assumption, in the universal cover ${\text {T}}^1({\mathbb {H}}^{d+1})$, the point ${{\mathcal {G}}}_n x$ is contained in a horoball $H_p(\eta )$. Hence $h_n<\eta h(p)$. If $h_n\le \eta h(p)/2$, then we can use Pythagorean’s theorem to conclude that $d(\pi (x),p)\le \sqrt{\eta h(p)h_n}$ (see Figure 4). If $h_n\ge \eta h(p)/2$, then $d(\pi (x),p)\le \eta h(p)/2\le \sqrt{\eta h(p)h_n}$. $\square$

Fig. 4

Radius

Lemma 6.38

Fix $c_{9}<\min \{1/C_{5},1/C_{8}^2\}$, where $C_{5}$ and $C_{8}$ are constants given in Lemma 6.13 and Lemma 6.32 respectively. For any $x\in \widetilde{\Omega _n}$, if ${\mathcal {G}}_n x\in {\mathcal {C}}_{c_{9}\eta }$, then $\pi (x)\in B_n$.

Proof

For $x\in \widetilde{\Omega _n}$, if ${\mathcal {G}}_n x\in {\mathcal {C}}_{c_{9}\eta }$, then it follows from Lemma 6.37 that there exists a parabolic fixed point p with $c_{9}\eta h(p)>h_n$ such that

$$\begin{aligned} d(\pi (x),p)<\sqrt{c_{9}\eta h(p) h_n}\le c_9\eta h(p). \end{aligned}$$

(6.39)

By the definition of $P_n$ and $Q_n$, this p must belong to $\bigcup _{j< n}(P_j\cup Q_j)$. If p is in some $P_j$, then by Lemma 6.13 we have

$$\begin{aligned} \eta h(p)/C_{5}<d(\pi (x),p), \end{aligned}$$

contradicting the assumption that $c_{9}<1/C_{5}$. So p must be in some $Q_j$. We use the construction of $B_n$, Lemma 6.32, that is $B_n\supset B(p,\sqrt{\eta h(p)h_n}/C_{8})$, and (6.39) to conclude that $\pi (x)\in B_n$. $\square$

Remark

If $\pi (x)\in B_n$, then the point ${{\mathcal {G}}}_nx$ is contained in ${{\mathcal {C}}}_{C\eta }$. So the set ${{\mathcal {G}}}_n \widetilde{B_n}$ is almost the same as the set of points in the cusps at time $t=n$, i.e. ${{\mathcal {C}}}_\eta \cap {{\mathcal {G}}}_n\widetilde{\Omega _{n}}$.

6.5.2 Parabolic fixed points, $B_n$ and different generations

Lemma 6.40

We have $P_{n}\cap (\cup _{l\le n} Q_l'')=\emptyset$.

Proof

If not, suppose $p\in P_{n}$ is also contained in an equivalence class $C(p')$ with $p'\in Q_m-(\cup _{1\le l\le m-1}Q_l'')$ and $m\le n$. Due to $\# C(p)\ge 2$, by the construction of equivalence classes, we must have $p'\in Q_m'$. Let $\partial$ be the associated boundary component of $p'$. Recall the construction of the equivalence classes. If $\partial$ is $\partial \Omega _0$, it is easy to obtain $d(p, \partial \Omega _0)<h_m$. If $\partial$ is some $\partial J_q$, due to $\# C(p')\ge 2$, we use Lemma 6.23 to deduce that $d(p,J_q)\le h_m$. By Lemma 6.27, we have $h_m/h_n\le h(p')/h(p)\le C_{7}$. Hence by the definition of $P_{n}$

$$\begin{aligned} d(p,\partial )\ge h_{n-1}/(4\eta )\ge h_m/(4eC_{7}\eta )> h_m, \end{aligned}$$

which is a contradiction. $\square$

Lemma 6.41

There exists a constant $0<c_{10}<1$ such that for any $p\in P_{n+1}\cup Q_{n+1}''$, we have

$$\begin{aligned} d(p,B_n)\ge c_{10}h_n/\eta . \end{aligned}$$

(6.42)

Proof

Let $p\in P_{n+1}\cup Q_{n+1}''$ and $B_{q,n}$ be a ball in $B_n$. By Lemma 6.40, p and q are two different parabolic fixed points. We have

$$\begin{aligned} d(p, B_{q,n})&\ge d(p,q)-C\sqrt{\eta h(q) h_n}\,\,\,\,\,\,\,\,\,\,\,\, (\text {by Lemma }~6.32)\\&\ge \sqrt{h(p)h(q)}-C\sqrt{\eta h(q) h_n}\,\,\,\,\,\,\,\,\,\,\,\, (\text {by Lemma }6.16)\\&=\sqrt{h(q)}(\sqrt{h(p)}-C\sqrt{\eta h_n})\\&\ge \sqrt{h_n/C\eta }\left( \sqrt{h_n/(Ce\eta )}-C\sqrt{\eta h_n}\right) \ge h_n/(C\eta ). \end{aligned}$$

$\square$

Recall that $D_{n+1}=\cup _{p\in P_{n+1}} J_p$ and $\Omega _{n+1}=\Omega _n-D_{n+1}$.

Lemma 6.43

If $\eta <\frac{c_{10}}{C_{5}C_{8}}$, then:

1. We have the followings:

$$\begin{aligned} D_{n+1}\cap B_n&=\emptyset \,\,\,\text {and}\,\,\, \left( \cup _{p\in Q''_{n+1}} B_{p,n+1}\right) \cap B_n=\emptyset , \nonumber \\ A_{n+1}&=(A_n-D_{n+1}-A_n\cap B_{n+1})\cup (A_{n+1}\cap B_n), \end{aligned}$$

(6.44)

$$\begin{aligned} A_n\cap B_{n+1}&=\cup _{p\in Q_{n+1}''}(B_{p,n+1}\cap \Omega _{n+1}),\ A_{n+1}\cap B_n=B_n-B_{n+1}. \end{aligned}$$

(6.45)

2. For $p\in Q_n''$ and $m\ge n$, we have $B_{p,m}\cap \Omega _m=B_{p,m}\cap \Omega _n$.

Proof

For $p\in P_{n+1}$, by Lemma 6.13, we have $J_p\subset B(p,C_{5}\eta h(p))$. Then $C_{5}\eta h(p)\le C_{5}h_n\le c_{10}h_n/\eta$. By Lemma 6.41, we have $J_p\cap B_n=\emptyset$. For $p\in Q_{n+1}''$, by Lemma 6.32, we have $B_{p,n+1}\subset B(p, C_{8}\sqrt{\eta h(p)h_n})$. Then $C_{8}\sqrt{\eta h(p)h_n}\le C_{8} h_n< c_{10}h_n/\eta$.

By Lemma 6.41, we have $B_{p,n+1}\cap B_n=\emptyset$.

The rest of the first statement can be obtained easily.

For $m>l\ge n$, by $D_{l+1}\cap B_l=\emptyset$ and $B_{p,m}\cap \Omega _l\subset B_{l}$ when $p\in Q_n''$ we know that

$$\begin{aligned} B_{p,m}\cap \Omega _{l+1}=B_{p,m}\cap (\Omega _{l}-D_{l+1})=B_{p,m}\cap \Omega _{l}, \end{aligned}$$

which implies the second part of the statement. $\square$

6.6 Energy exchange argument

We are ready to prove Proposition 6.15.

Lemma 6.46

There exists $c_{11}>0$ such that

$$\begin{aligned} \mu (B_{n}\cap A_{n+1})>c_{11}\mu (B_{n}). \end{aligned}$$

(6.47)

The definition of equivalence classes is mainly used in this lemma. The idea is that the left hand side of (6.47) can be expressed as a sum over equivalence classes. Over one equivalence class, we obtain a full ball whose measure we are able to estimate (Fig. 5).

Proof of Lemma 6.46

We claim that for any distinct $p,p'\in \cup _{1\le l\le n}Q''_l$, we have $B_{p,n}\cap B_{p',n}=\emptyset$. The first equation in Lemma 6.43 verifies the case when $p\in Q''_l$ and $p'\in Q''_j$ with $l\ne j$.

When $p,p'\in Q''_l$, using Lemma 6.16, and 6.32, we have

$$\begin{aligned}&d(B_{p,l}, B_{p',l})\ge d(p,p')-C\sqrt{\eta h(p) h_l}-C\sqrt{\eta h(p')h_l}\\&\quad \ge \sqrt{h(p) h(p')}-C\sqrt{\eta h(p) h_l}-C\sqrt{\eta h(p')h_l}\\&\quad \ge h_l/(C\eta ) -2Ch_{l-1}>0, \end{aligned}$$

showing the claim.

So $\mu (B_n\cap A_{n+1})$ can be divided into the sum over $p\in \cup _{1\le l\le n}Q''_l$. Using Lemma 6.35, we can further group the sum into equivalence classes. Due to (6.45), $\mu (B_n\cap A_{n+1})=\mu (B_n-B_{n+1})$. Then the proof of (6.47) is reduced to proving that there exists $c_{11}>0$ such that for each equivalence class C(p), we have

$$\begin{aligned}&\sum _{p'\in C(p)} \mu ((B_{p',n}-B_{p',n+1})\cap \Omega _n)\ge c_{11} \sum _{p'\in C(p)}\mu (B_{p',n}\cap \Omega _n). \end{aligned}$$

We first consider the equivalence classes defined in Case 1 and Case 3 in page 28. Then by the definition of equivalence class and Lemma 5.18, we obtain

$$\begin{aligned}&\frac{\sum _{p'\in C(p)} \mu ((B_{p',n}-B_{p',n+1})\cap \Omega _n)}{\sum _{p'\in C(p)}\mu (B_{p',n}\cap \Omega _n)}\\&\quad \ge \frac{\mu (B_{p,n}-B_{p,n+1})}{C\mu (B_{p,n})}=\frac{\mu (B(p,r_{p,n})-B(p,r_{p,n+1}))}{C\mu (B(p,r_{p,n}))}\ge \frac{c}{C}, \end{aligned}$$

where the last inequality follows from Lemma 5.21 and $r_{p,n}=\sqrt{\eta h_mh_p}\ll \eta h(p)$.

Fig. 5

Pairing partial balls: $q=\gamma ^{-1}p_i$, $p'=g_i\gamma p$, $B_1=B(g_i\gamma p, {{\tilde{r}}}_{p,m}/C_{6})$, $\gamma _1\in (g_i\Gamma g_i^{-1})_{\infty }$

Next we consider the equivalence classes defined in Case 2 (Fig. 5). Suppose the associated boundary component of p is $\partial J_{q}$ with $q=\gamma ^{-1}p_i$ and $\gamma ^{-1}$ is the representation of q. We first assume that $p_i=\infty$. By Lemma 5.17 and (6.30) for any Borel subset $E\subset B_{p,n}$, we have

$$\begin{aligned} h^{\delta }_q\mu (\gamma E)/C\le \mu (E)\le Ch^{\delta }_q\mu (\gamma E). \end{aligned}$$

We have

$$\begin{aligned}&\frac{\sum _{p'\in C(p)} \mu ((B_{p',n}-B_{p',n+1})\cap \Omega _n)}{\sum _{p'\in C(p)}\mu (B_{p',n}\cap \Omega _n)}\nonumber \\&\quad \ge \frac{\sum _{p'\in C(p)} \mu ((B (\gamma p',r_{p',n})-B(\gamma p',r_{p',n+1}))\cap \gamma J_q^c)}{C\sum _{p'\in C(p)}\mu (B (\gamma p',r_{p',n})\cap \gamma J_q^c)}. \end{aligned}$$

(6.48)

For each $p'\in C(p)$, we can write $p'=\gamma ^{-1} \gamma _1 \gamma p$ with $\gamma _1\in \Gamma _{\infty }$. We have

$$\begin{aligned}&\mu (B (\gamma p',r_{p',n})\cap \gamma J_q^c)=\mu (\gamma _1 B (\gamma p, r_{p,n})\cap \gamma J_q^c)\\&\quad \approx \mu (B (\gamma p, r_{p,n})\cap \gamma ^{-1}_1\gamma J_q^c), \end{aligned}$$

where we use Lemma 5.18 and (6.29) to compute $|\gamma _1(x)|$ and |x| for x in $B (\gamma p, r_{p,n})$. Summing over $p'\in C(p)$, we can get a full ball. Similarly, we have

$$\begin{aligned}&\mu ((B (\gamma p', r_{p',n})-B (\gamma p',r_{p',n+1}))\cap \gamma J_q^c)\\&\quad \approx \mu ((B(\gamma p, r_{p,n})-B(\gamma p,r_{p,n+1}))\cap \gamma _1^{-1}\gamma J_q^c). \end{aligned}$$

We use these two observations and Lemma 5.21, $r_{p,n}\ll \eta h(\gamma p)$ (6.22) to conclude

$$\begin{aligned} (6.48)\ge&\frac{\mu (B(\gamma p, r_{p,n})-B(\gamma p, r_{p,n+1}))}{C\mu (B (\gamma p, r_{p,n}))}\ge \frac{c}{C}. \end{aligned}$$

For general $p_i$, let $g_ip_i=\infty$ and $\Gamma _i=g_i\Gamma g_i^{-1}$. Using (5.13), we obtain

$$\begin{aligned}&\frac{\sum _{p'\in C(p)} \mu ((B_{p',n}-B_{p',n+1})\cap \Omega _n)}{\sum _{p'\in C(p)} \mu (B_{p',n}\cap \Omega _n)}\nonumber \\&\quad \quad \approx \frac{\sum _{p'\in C(p)} \mu _{\Gamma _i}(g_i(B_{p',n}-B_{p',n+1})\cap g_i\Omega _n)}{\sum _{p'\in C(p)} \mu _{\Gamma _i} (g_iB_{p',n}\cap g_i\Omega _n)}. \end{aligned}$$

(6.49)

This fraction can be estimated the same way as we estimate (6.48). So

$$\begin{aligned} (6.49)\ge c/C. \end{aligned}$$

$\square$

Let $C_{12}=2C_{5} C_{3}+4C_{8}$, where $C_{3}$, $C_{5}$ and $C_{8}$ are constants given by Proposition 5.14, Lemma 6.13 and Lemma 6.32 respectively. Let

$$\begin{aligned} \Omega _n'=\{x\in \Omega _n:\ d(x,\partial \Omega _n)\le C_{12}h_n \}. \end{aligned}$$

This is the set of points with distance less than $C_{12} h_n$ to the boundary of $\Omega _{n}$.

Lemma 6.50

(Boundary estimate) There exists $c_{13}>0$ depending on $C_{12}\eta$ such that

$$\begin{aligned} \mu (\Omega _n')\le c_{13}\mu (\Omega _n) \end{aligned}$$

and $c_{13}$ tends to 0 as $C_{12}\eta$ tends to 0.

Proof

The boundary $\partial \Omega _n$ consists of $\partial \Omega _0$ and $\partial J_p$ with $p\in \cup _{1\le l\le n}P_l$. For any $p\in \cup _{1\le l\le n}P_l$, write $p=\gamma p_i$ with $\gamma \in \Gamma$ the representation of p and $\Gamma _i=g_i\Gamma g_i^{-1}$. Recall the definitions (5.28) and (6.8). Note that $h_{n}/(4\eta )\le h(p)$. It follows from Lemmas 5.11 and 5.12 that there exists $C>1$ such that $h_{n}/(C\eta )<h(g_i p)$ and

$$\begin{aligned} g_iN_{C_{12}h_n}(\partial J_p)\subset N_{CC_{12} h_n}(\partial J_{i,p}),\,\,\, N_{h_n/(C\eta )}(\partial J_{i,p})\subset g_i N_{h_n/(4\eta )}(\partial J_p), \end{aligned}$$

where $J_{i,p}$ is defined as in (6.11). It follows from (5.13), Lemma 5.29 and 6.9 that there exists $c>0$ such that

$$\begin{aligned} \mu (\Omega _n')&=\mu (N_{C_{12} h_n}(\partial {\Omega _0}))+\sum _{p\in \cup _{1\le l\le n}P_l}\mu (N_{C_{12}h_n}(\partial J_p))\\&\le c\mu (N_{h_n/(4\eta )}(\partial \Omega _0))+C'\sum _{p\in \cup _{1\le l\le n}P_l}\mu _{\Gamma _i}(N_{CC_{12}h_n}(\partial J_{i,p}))\\&\le c\mu (N_{h_n/(4\eta )}(\partial \Omega _0))+cC'\sum _{p\in \cup _{1\le l\le n}P_l}\mu _{\Gamma _i}(N_{h_n/(C\eta )}(\partial J_{i,p}))\\&\le c\mu (N_{h_n/(4\eta )}(\partial \Omega _0))+cC'^2\sum _{p\in \cup _{1\le l\le n}P_l}\mu (N_{h_n/(4\eta )}(\partial J_p))\\&\le cC'^2\mu (\Omega _n), \end{aligned}$$

where the last inequality is due to Lemma 6.17 and $C'=\max _{i}e^{\delta d(o,g_io)}$. $\square$

Lemma 6.51

There exists $0<c_{14}<1$ such that

$$\begin{aligned}\mu (A_n\cap (D_{n+1}\cup B_{n+1} ))\le c_{14}\mu (A_n)+\mu (\Omega _n'). \end{aligned}$$

Proof

By Lemma 6.43, we have

$$\begin{aligned} A_n\cap B_{n+1}\cap (\Omega _n-\Omega _n')\subset \bigcup _{p\in Q_{n+1}''}B_{p,n+1}. \end{aligned}$$

We consider the points $p\in Q_{n+1}''$ such that $B_{p,n+1}$ intersects the set on the left. Denote the set of such points by $Q_{n+1}'''$. By Lemma 6.32, we have

$$\begin{aligned} B_{p,n+1}\subset B(p,C_{8}\sqrt{\eta h(p)h_{n+1}})\subset B(p,C_{8}h_n). \end{aligned}$$

Then its distance to $\partial \Omega _n$ is greater than $(C_{12}-2C_{8})h_n\ge C_{12}h_n/2$. So $Q_{n+1}'''$ must be a subset of points in Case 3 in page 28, and $B_{p,n+1}=B(p,\sqrt{\eta h(p)h_{n+1}})\subset B(p,h_n)$.

For $p\in P_{n+1}$, by Lemma 6.13, we have $J_p\subset B(p,C_{5}\eta h(p))\subset B(p,C_{5}h_n)$.

By (6.42), for $p\in P_{n+1}\cup Q_{n+1}'''$

$$\begin{aligned} d(p,B_n)\ge c_{10}h_n/\eta \ge C_{12}h_n/2. \end{aligned}$$

(6.52)

Hence

$$\begin{aligned} B(p, C_{5}h_{n})\subset B(p,C_{12} h_n/2)\subset A_n. \end{aligned}$$

Then by doubling property in Proposition 5.14

$$\begin{aligned}\mu (B(p, C_{5}h_{n}))\le c_{14}\mu (B(p,C_{12} h_n/2)). \end{aligned}$$

By Lemma 6.16, the points in the set $P_{n+1}\cup Q_{n+1}'''$ are of distance $h_{n+1}/\eta$ apart from each other. Hence the balls $B(p,C_{12}h_n/2)$ are disjoint. Adding them together, we obtain

$$\begin{aligned}&\mu (A_n\cap (D_{n+1}\cup B_{n+1})-\Omega _n')\le \sum _{p\in P_{n+1}\cup Q_{n+1}'''}\mu (B(p,C_{5}h_n))\\&\qquad \le c_{14} \sum _{p\in P_{n+1}\cup Q_{n+1}'''}\mu (B(p,C_{12}h_n/2))\le c_{14}\mu (A_n). \end{aligned}$$

$\square$

Set $A_n'=A_n-\Omega _n'$ which is the set of points in $A_n$ with distance at least $C_{12}h_n$ to the boundary $\partial \Omega _n$.

Lemma 6.53

There exist $N\in {\mathbb {N}}$ and $c_{15}>0$ depending on $\eta$ such that

$$\begin{aligned} \mu (\cup _{l=1}^N D_{n+l})\ge c_{15}\mu (A_n'). \end{aligned}$$

Let $\widetilde{A_n}$ be the subset of $\widetilde{\Omega _n}$ such that $\pi (\widetilde{A_n})=A_n$. The key point of the proof is that we can use the recurrence property of the geodesic flow on ${{\mathcal {G}}}_n(\widetilde{A_n})$, since Lemma 6.38 tells us that ${{\mathcal {G}}}_n(\widetilde{A_n})$ stays in a compact subset. Recall that we introduce some notations. We assumed that there are j cusps in M and $\{p_i\}_{1\le i\le j}$ is a complete set of inequivalent parabolic fixed points. We used the notation $H_{p_i}$ to denote the horoball based at $p_i$. Now let $H^s_{p_i}\subset {\text {T}}^1({\mathbb {H}}^{d+1})$ be the strong stable horosphere, that is,

$$\begin{aligned} H_{p_i}^s:=\{x\in {\text {T}}^1(H_{p_i}):\,\text {the basepoint of}\,x\,\text {is at}\,\partial H_{p_i}\,\text {and}\,\pi (x)=p_i\}. \end{aligned}$$

By abusing the notation, we also use $H_{p_i}^s$ to denote its image in the quotient space ${\text {T}}^1(M)$.

For every $x\in {\text {T}}^1(M)$ and $\epsilon >0$, set $W^u(x,\epsilon )$ to be the local strong unstable manifold at x, that is,

$$\begin{aligned} W^u(x,\epsilon ):=\{y\in {\text {T}}^1(M):\,\lim _{t\rightarrow -\infty }d({\mathcal {G}}_{t}x,{\mathcal {G}}_ty)=0,\ d^u(y,x)\le \epsilon \}, \end{aligned}$$

where $d(\cdot ,\cdot )$ is the Riemannian metric on ${\text {T}}^1(M)$ and $d^u(\cdot ,\cdot )$ is the Riemannian metric restricted on the strong unstable manifold.

Denote by W in ${\text {T}}^1(M)$ the non-wandering set of the geodesic flow.

Lemma 6.54

Let K be any compact subset in W. Then there exists $U_0>0$ such that for every x in K and every $H^s_{p_i}$ in ${\text {T}}^1(M)$, there exists a time $t\in [0,U_0]$ such that ${{\mathcal {G}}}_t(W^u(x,1))$ meets $H^s_{p_i}$.

Proof

Let $\epsilon <1/10$ and consider $Z_1=\cup _{x\in \partial H^s_{p_i}}W^u(x,\epsilon )$ and $Z_2=\cup _{x\in \partial H^s_{p_i}}W^u(x,5\epsilon )$ in ${\text {T}}^1(M)$. Then $Z_1$ is a transversal section to the geodesic flow. By ergodicity of geodesic flow on non-wandering set W, there exists a point y such that its negative time orbit is dense and there exists $t_0\ge 0$ such that ${{\mathcal {G}}}_{t_0} y\in Z_1$. We can cover the compact set K with a finite number of balls of radius $\epsilon$. There exists $t_1>0$ such that ${{\mathcal {G}}}_{[-t_1,0]}y$ intersects every ball.

Fix any x in K. There exists $x'\in W^u(x,\epsilon )$ and $-s\in [-t_1,0]$ such that $d(x',{\mathcal {G}}_{-s}y)\le 2\epsilon$ and ${{\mathcal {G}}}_{-s}y$ are in the same strong stable manifold (that is to say, $\lim _{t\rightarrow \infty }d({\mathcal {G}}_t x', {\mathcal {G}}_t ({\mathcal {G}}_{-s}y))=0$). Therefore

$$\begin{aligned} d({{\mathcal {G}}}_{s+t_0}x',{{\mathcal {G}}}_{t_0} y)\le 2\epsilon . \end{aligned}$$

Using ${{\mathcal {G}}}_{t_0} y\in Z_1$ and local product structure, we have ${{\mathcal {G}}}_{s+t_0}x'\in {{\mathcal {G}}}_{s_1}Z_2$ for some $s_1\in [-\epsilon ,\epsilon ]$. Due to the definition of $Z_2$, we can find $x''\in W^u(x,6\epsilon )$ such that ${{\mathcal {G}}}_{s+t-s_1}x''\in H^s_{p_i}$. $\square$

The following lemma is a straightforward corollary of Lemma 6.54. Recall that $c_{9}>0$ is the constant given in Lemma 6.38. Let $K_{c_{9}\eta }=W-\Gamma \backslash {{\mathcal {C}}}_{c_{9}\eta }$. The base of non-wandering set in M is the convex core C(M) and the base of $\Gamma \backslash {{\mathcal {C}}}_{c_{9}\eta }$ is a union of proper horocusps. By Definition 2.2, we know that $K_{c_{9}\eta }$ is compact.

Lemma 6.55

Let $U_0$ be the constant in Lemma 6.54 with $K=K_{c_{9}\eta }$. For every x in $\Delta _0\cap \Lambda _\Gamma$ and $n\in {\mathbb {N}}$, if ${{\mathcal {G}}}_n {\tilde{x}}$ is in $K_{c_{9}\eta }$, where ${\tilde{x}}$ is the point in $\widetilde{\Omega _0}$ such that $\pi ({\tilde{x}})=x$, then the ball $B(x,h_n)$ contains a parabolic fixed point with height in $h_n[e^{-U_0},1]$.

Proof

Let ${\tilde{B}}$ be the set in $\widetilde{\Omega _0}$ such that $\pi ({\tilde{B}})=B(x,h_n)$. We have ${{\mathcal {G}}}_n{\tilde{B}}=W^u({{\mathcal {G}}}_n{\tilde{x}},1)$. As ${{\mathcal {G}}}_n{\tilde{x}}\in K_{c_{9}\eta }$, by Lemma 6.54, there exists $t\in [0,U_0]$ such that ${{\mathcal {G}}}_tW^u({{\mathcal {G}}}_n{\tilde{x}},1)$ intersects some $H^s_{p_i}$. Hence in the universal cover ${\text {T}}^1({\mathbb {H}}^{d+1})$, the unstable leaf ${{\mathcal {G}}}_tW^u({{\mathcal {G}}}_n{\tilde{x}},1)$ is tangent to a horoball. Let q be the basepoint of the horoball. Then q is in $B(x,h_n)$. $\square$

Proof of Lemma 6.53

Set $N=U_0+2\lfloor -\log \eta \rfloor +2$. We claim that: There exists $C'>1$ depending on $\eta$ such that $\mathop {\cup }_{1\le l\le N}P_{n+l}$ is a $C'h_n$ dense set in $A_n'\cap \Lambda _\Gamma$. That is to say, for every $x\in A_n'\cap \Lambda _\Gamma$, there exists some $p\in \bigcup _{1\le l\le N}P_{n+l}$ such that $d(x,p)\le C'h_n$.

Let $k=\lfloor -\log \eta \rfloor$. Fix any point $x\in A_n'\cap \Lambda _\Gamma$. We consider the position of x in $\Omega _{n+k}$.

• Case 1 Suppose $x\notin \Omega _{n+k}$. Then $x\in J_{p}$ for some $p\in \cup _{1\le l\le k}P_{n+l}$. So $d(x,p)\le C_{5}\eta h(p)\le C_{5}h_n$.

• Case 2 Suppose $x\in \Omega _{n+k}$ and $d(x,\partial \Omega _{n+k})<3h_{n+k}/\eta$. As $x\notin \Omega _n'$, we have $d(x,\partial \Omega _n)\ge C_{12}h_n$. Meanwhile, we have $3h_{n+k}/\eta < C_{12} h_n$. Consequently, the connected component in $\partial \Omega _{n+k}$ closest to x is some $\partial J_p$ with $p\in \cup _{1\le l\le k} P_{n+l}$. Hence

  $$\begin{aligned} d(x,p)\le d(x,\partial J_p)+d(\partial J_p,p)\le 3h_{n+k}/\eta +C_{5} \eta h(p)\le Ch_n. \end{aligned}$$

• Case 3 Suppose $x\in A_{n+k}\cap \Lambda _\Gamma$ and $d(x,\partial \Omega _{n+k})>2h_{n+k}/\eta$. By Lemma 6.38, ${\mathcal {G}}_{n+k}{{\tilde{x}}}\in K_{c_{9}\eta }$. It then follows from Lemma 6.55 that $B(x, h_{n+k})$ contains a parabolic fixed point p with height in $h_{n+k}[e^{-U_0},1]$. Let $j=\lfloor -\log (\eta h(p))\rfloor$, then $j\in n+2k+[0,U_0+1]$. Let’s consider the position of p.

  • Suppose $p\in P_{j+1}$. Then $d(x,p)\le h_{n+k}$.

  • Suppose $p\notin P_{j+1}$ and $p\notin \Omega _j$. Note that the conditions on x and $p\in B(x,h_{n+k})$ imply that $p\in \Omega _{n+k}$. So there exists some $q\in \cup _{l=1}^{j-n-k}P_{n+k+j}$ such that $p\in J_q$. We obtain

    $$\begin{aligned} d(x,q)\le d(x,p)+d(p,q)\le h_{n+k}+C_{5}\eta h(q)\le C h_{n+k}. \end{aligned}$$

  • Suppose $p\notin P_{j+1}$ and $p\in \Omega _j$. Because $\eta h(p)\in (h_{j+1},h_j]$, we must have $p\in Q_{j+1}$. By the definition of $Q_{j+1}$, we have $d(p,\partial \Omega _j)\le h_j/\eta$. Observe that

    $$\begin{aligned} d(p,\partial \Omega _{n+k})\ge d(x,\partial \Omega _{n+k})-d(x,p)>2h_{n+k}/\eta -h_{n+k}>h_j/\eta . \end{aligned}$$

    So there exists $q\in \cup _{l=1}^{j-n-k}P_{n+k+l}$ such that $d(p,J_{q,\eta })\le h_j/\eta$. This implies

    $$\begin{aligned} d(x,q)\le d(x,p)+d(p,q)\le h_{n+k}+h_j/\eta +C_{5}\eta h(q)\le Ch_{n+k}. \end{aligned}$$

• Case 4 Suppose $x\in B_{n+k}$ and $d(x,\partial \Omega _{n+k})\ge 3h_{n+k}/\eta$. As $x\in A_n$, we have $x\in B_{n+k}-B_n$. So there exists $p\in \cup _{1\le l \le k}Q_{n+l}''$ such that $x\in B_{p, n+k}$. By Lemma 6.32, we have

  $$\begin{aligned} B_{p,n+k}\subset B(p,C_{8}\sqrt{\eta h(p) h_{n+k}})\subset B(p,C_{8}\sqrt{h_n h_{n+k}}). \end{aligned}$$

  Since $h_{k}\ge \eta$, for any $y\in B_{p,n+k}$, using Lemma 6.32, we have

  $$\begin{aligned} d(y,\partial \Omega _{n+k})\ge d(x,\partial \Omega _{n+k})-d(x,y)\ge 3h_{n+k}/\eta -2C_{8}\sqrt{h_n h_{n+k}}\ge 2h_{n+k}/\eta . \end{aligned}$$

  So the full ball $B_{p,n+k}$ is contained in $\Omega _{n+k}$. By a similar argument as in the proof of Lemma 6.46, we have $\mu (B_{p,n+k}-B_{p,n+k+1})>0$. We can find a point $y\in \Lambda _{\Gamma }\cap (B_{p,n+k}-B_{p,n+k+1})$. By (6.45), we know that in fact y is in $A_{n+k+1}\cap \Lambda _{\Gamma }$.

  • If $d(y,\partial \Omega _{n+k+1})>2h_{n+k+1}/\eta$, the point y belongs to Case 3.

  • Otherwise, $d(y,\partial \Omega _{n+k+1})\le 2h_{n+k+1}/\eta$. But $d(y,\partial \Omega _{n+k})>2h_{n+k}/\eta$, there exists $J_q$ with $q\in P_{n+k+1}$ such that $d(y,\partial J_q)\le 2h_{n+k+1}/\eta$.

  It follows that there exists $q\in \cup _{1\le l\le N}P_{n+l}$ such that

  $$\begin{aligned} d(x,q)\le d(x,y)+d(y,q)\le 2C_{8}\sqrt{h_{n}h_{n+k}}+d(y,q)\le Ch_n. \end{aligned}$$

Finally, by Lemma 6.13 we know that for $p\in \cup _{1\le l\le N}P_{n+l}$, the balls $B(p,\eta h(p)/C_{5})$ are disjoint. Using the claim and doubling property (Proposition 5.14),

$$\begin{aligned}&\mu (\cup _{l=1}^N D_{n+l})\ge \sum _{p\in \cup _{1\le l\le N}P_{n+l}}\mu (B(p,\eta h(p)/C_{5}))\\&\quad \ge c_{15}\sum _{p\in \cup _{1\le l\le N}P_{n+l}}\mu (B(p,C' h_n ))\ge c_{15}\mu (A_n'\cap \Lambda _\Gamma )=c_{15}\mu (A_n'), \end{aligned}$$

finishing the proof. $\square$

Proof of Proposition 6.15

We will prove the following statement and Proposition 6.15 is a direct consequence of this: for $\eta$ sufficiently small, there exist N and $c_0>0$ depending on $\eta$ such that

$$\begin{aligned} \mu (\cup _{l=1}^N D_{n+l})\ge c_0\mu (\Omega _n). \end{aligned}$$

Recall that $c_{11},c_{13}$ and $c_{14}$ are the constants given in Lemma 6.46, 6.50 and 6.51 respectively. We can take $c_{13}$ small enough such that $c_{13}<c_{11}$ and $c_{13}+c_{14}<1$. Write $t_n=\frac{\mu (A_n)}{\mu (B_n)}$, which makes sense even if $\mu (B_n)=0$. Then by Lemma 6.43, 6.46, 6.51, and 6.50

$$\begin{aligned} t_{n+1}&=\frac{\mu (A_{n+1})}{\mu (B_{n+1})}=\frac{\mu (A_n)+\mu (B_n\cap A_{n+1})-\mu (A_n\cap (D_{n+1}\cup B_{n+1}))}{\mu (B_n)-\mu (B_n\cap A_{n+1})+\mu (A_n\cap B_{n+1})}\\&\ge \frac{\mu (A_n)+c_{11}\mu (B_n)-(c_{14}\mu (A_n)+c_{13}\mu (\Omega _n))}{\mu (B_n)-c_{11}\mu (B_n)+(c_{14}\mu (A_n)+c_{13}\mu (\Omega _n))}\\&=\frac{t_n-(c_{14}+c_{13})t_n+(c_{11}-c_{13})}{1+(c_{14}+c_{13})t_n-(c_{11}-c_{13})}=f(t_n). \end{aligned}$$

Here f is fractional function and of the form $f(t)=\frac{a_1t+a_2}{b_1t+b_2}$ with $a_i,b_i>0$, which is a monotone function. Hence

$$\begin{aligned} \inf _{t\in {\mathbb {R}}^+ }f(t)\ge & {} \min \left\{ \frac{a_1}{b_1},\ \frac{a_2}{b_2} \right\} \\= & {} \min \left\{ \frac{1-(c_{14}+c_{13})}{c_{14}+c_{13}},\frac{c_{11}-c_{13}}{1-(c_{11}-c_{13})} \right\} =q(c_{13}). \end{aligned}$$

By $t_n>0$, there is a uniform lower bound of $t_n$ for all $n\in {\mathbb {N}}$.

Then use Lemma 6.53 to obtain the desired statement:

$$\begin{aligned}&\mu (\cup _{l=1}^N D_{n+l})\ge c_{15}\mu (A_n')\ge c_{15}(\mu (A_n)-\mu (\Omega _n'))\\&\quad \ge c_{15}(\mu (A_n)-c_{13}\mu (\Omega _n))=c_{15}\left( \frac{t_n}{1+t_n}-c_{13}\right) \mu (\Omega _n). \end{aligned}$$

If $c_{13}$ is small enough, then $\frac{t_n}{1+t_n}\ge \frac{q(c_{13})}{1+q(c_{13})}>c_{13}$. Then we can fix a small $\eta$ in Lemma 6.50 such that $c_{13}$ satisfies these restriction. $\square$

6.7 Exponential tail

For one cusp case, we have described how to construct the countable collection of disjoint open subsets in $\Delta _0$ and the expanding map in Sect. 6.2. When there are multi-cusps, the coding is constructed in two steps and we describe the first step here and finish the rest in Sect. 6.8.

Suppose that there are j cusps. Recall the regions $\Delta _{p_i}$ introduced in Sect. 5.2. We claim: there is a countable collection of disjoint open subsets $\sqcup _{i,k}\Delta _{p_i,k}$ in $\Delta (=\sqcup _i \Delta _{p_i})$ and an expanding map $T_0:\sqcup _{i,k}\Delta _{p_i,k}\rightarrow \Delta$ such that

• $\sum _{i,k}\mu (\Delta _{p_i,k})=\mu (\Delta )$.

• For each $\Delta _{p_i,k}$, it is a subset in $\Delta _{p_i}$. And there exists an element $\gamma _0\in \Gamma$ such that $\Delta _{p_i,k}=\gamma _0 \Delta _{p_l}$ for some $1\le l\le j$ and $T|_{\Delta _{p_i,k}}=\gamma _0^{-1}$.

Denote by ${\mathcal {H}}_0$ the set of inverse branches of $T_0$.

The construction is as follows: for each $\Delta _{p_i}$, we apply the construction in Sect. 6.2 to the group $\Gamma _i=g_i\Gamma g_i^{-1}$ and the region $g_i\Delta _{p_i}=\Delta _{p_i}'$. In particular, we attain a countable collection of disjoint open sets $\Delta _{p_i,k}'$. Moreover, Proposition 6.15 holds for $\Delta _{p_i}'$. We set $\Delta _{p_i,k}=g_i^{-1}\Delta _{p_i,k}'$.

For an element $\gamma _0$ in ${{\mathcal {H}}}_0$, if $\gamma _0$ maps $\Delta _{p_l}$ into some $\Delta _{p_i,k}$, then we define

$$\begin{aligned} |\gamma _0'|_\infty =\sup _{x\in \Delta _{p_l}}|\gamma _0'(x)|. \end{aligned}$$

(6.56)

The infinity norm of the derivative of a composition map is defined similarly. We prove the following.

Lemma 6.57

There exists $\epsilon >0$ such that

$$\begin{aligned} \sum _{\gamma _0\in {{\mathcal {H}}}_0} |\gamma _0'|_{\infty }^{\delta -\epsilon }<\infty . \end{aligned}$$

(6.58)

For one cusp case, this gives the exponential tail (4.2). When there are multi-cusps, (6.58) can be understood as that the map $T_0$ satisfies the exponential tail property.

We start the proof of Lemma 6.57 with the following result. Denote by $\cup _n P_n$ the set of “good parabolic fixed points" which appear in the first step of the construction of the coding for multi-cusp case and are defined similarly as (6.14).

Lemma 6.59

There exists $C>0$ such that for any parabolic fixed point $p=\gamma p_i\in \Delta _0\cap \cup _n P_n$, we have for any $\epsilon \in (0,\delta -k/2)$,

$$\begin{aligned} \sum _{\gamma _1\in N_p}|(\gamma \gamma _1)'|_{\infty }^{\delta -\epsilon }\le C(2\delta -k-2\epsilon )^{-1}h(p)^{-\epsilon } \eta ^{-2\epsilon }\mu (J_{p}), \end{aligned}$$

where k is the rank of the parabolic fixed point p and $N_p$ is defined in (6.4).

Proof

We first consider the case when $p=\gamma \infty$. By Lemma 5.7, we have for every $x\in \Delta _0$ and every $\gamma _1\in N_p$,

$$\begin{aligned} |(\gamma \gamma _1)'(x)|=|\gamma '(\gamma _1x)|=\frac{h(p)}{d(\gamma _1x,x_{\gamma })^2}. \end{aligned}$$

As $\cup _{\gamma _1\in N_p}\gamma _1\Delta _0\subset B(x_{\gamma }, 1/\eta )^c$ where $x_{\gamma }=\gamma ^{-1}\infty$, we use general polar coordinates to obtain

$$\begin{aligned} \sum _{\gamma _1\in N_p}|(\gamma \gamma _1)'|^{\delta -\epsilon }_{\infty }\ll h(p)^{\delta -\epsilon }\sum _{\gamma _1\in N_p}\frac{1}{d(\gamma _1\Delta _0,x_{\gamma })^{2\delta -2\epsilon }}\ll \frac{h(p)^{\delta -\epsilon } \eta ^{2\delta -2\epsilon -k}}{2\delta -2\epsilon -k}. \end{aligned}$$

(6.60)

Meanwhile, by the quasi-invariance of PS measure and (2.1), we have for every $\gamma _1\in N_p$

$$\begin{aligned} \mu (\gamma \gamma _1 \Delta _0)= & {} \int _{x\in \Delta _0}|(\gamma \gamma _1)'(x)|^{\delta }_{{\mathbb {S}}^n}\mathrm {d}\mu (x)\approx \int _{x\in \Delta _0}|(\gamma \gamma _1)'(x)|^{\delta }\mathrm {d}\mu (x)\\\approx & {} \frac{\mu (\Delta _0)h(p)^{\delta }}{d(\gamma _1\Delta _0,x_{\gamma })^{2\delta }}. \end{aligned}$$

Therefore,

$$\begin{aligned} \mu (\cup _{\gamma _1\in N_p} \gamma \gamma _1\Delta _0)\gg \mu (\Delta _0)h(p)^{\delta }\sum _{\gamma _1\in N_p}\frac{1}{d(\gamma _1\Delta _0,x_{\gamma })^{2\delta }}\gg \frac{h(p)^{\delta }\eta ^{2\delta -k}}{2\delta -k}. \end{aligned}$$

(6.61)

Hence (6.60) and (6.61) together yield the statement for the case when $p=\gamma \infty$.

For the general case when $p=\gamma p_i$ with $g_i p_i=\infty$. Note that for every $\gamma _1\in N_p$, we have $\gamma \gamma _1=g_i^{-1}(g_i\gamma \gamma _1 g_i^{-1}) g_i$. Hence by Lemma 5.11 and 5.12

$$\begin{aligned}&h(g_i p)\approx h(p),\ \ |(\gamma \gamma _1)'|_{\infty }=\sup _{x\in \Delta _{p_i}}|(\gamma \gamma _1)'(x)|\\&\quad \qquad \approx \sup _{x\in g_i\Delta _{p_i}}|(g_i \gamma \gamma _1 g_i^{-1})'(x)|= |(g_i \gamma \gamma _1 g_i^{-1})'|_{\infty }. \end{aligned}$$

Write $\Gamma _i=g_i\Gamma g_i^{-1}$. Using (5.13), we obtain

$$\begin{aligned} \mu (\gamma \gamma _1 \Delta _{p_i})\approx \mu _{\Gamma _i}(g_i\gamma \gamma _1g_i^{-1}(g_i\Delta _{p_i})). \end{aligned}$$

We have $g_ix_p=(g_i\gamma g_i^{-1})^{-1}\infty \in g_i{\overline{\Delta }}_{p_i}$. Because $g_ip=g_i\gamma g_i^{-1}\infty$ and $g_i \gamma _1 g_i^{-1}(g_i\Delta _{p_i})\subset B(g_ix_p,1/\eta )^c$ for any $\gamma _1\in N_p$, we are able to compare $\sum _{\gamma _1} |(g_i \gamma \gamma _1 g_{i}^{-1})'|^{\delta -\epsilon }_{\infty }$ with $\mu _{\Gamma _i}(\cup _{\gamma _1} g_i \gamma \gamma _1 g_i^{-1}(g_i \Delta _{p_i}))$ as above and this will prove Lemma 6.59 for the general case. $\square$

Proof of Lemma 6.57

We only need to sum the inverse branches in ${\mathcal {H}}_0$ whose images are in $\Delta _{0}$. For a general inverse branches whose image is in $\Delta _{p_j}$, we consider the group $g_j\Gamma g_j^{-1}$ and the inequality can be proved in the same fashion. By Lemma 6.59 and Proposition 6.15, for any sufficiently small $\epsilon \in (0,1)$,

$$\begin{aligned}&\sum _{n\in {\mathbb {N}}}\sum _{p=\gamma p_i\in P_n\cap \Delta _0}\sum _{\gamma _1\in N_p}|(\gamma \gamma _1)'|_{\infty }^{\delta -\epsilon } \ll \sum _{n\in {\mathbb {N}}}\sum _{p\in P_n}\mu (J_{p})h(p)^{-\epsilon }\eta ^{-2\epsilon }\\&\qquad \le \eta ^{-2\epsilon }\sum _{n\in {\mathbb {N}}}\mu (\Omega _n)e^{\epsilon (n+1)}\\&\qquad \le \eta ^{-2\epsilon }\sum _{n\ge N}(1-\epsilon _0)^ne^{\epsilon (n+1)}+\eta ^{-2\epsilon }\sum _{n< N}\mu (\Omega _n)e^{\epsilon (n+1)}. \end{aligned}$$

By choosing an $\epsilon$ small enough such that $(1-\epsilon _0)e^{\epsilon }<1$, the above sum is finite. $\square$

6.8 Coding for multi-cusps

We caution the readers that the symbol $\gamma$ will be used to denote an inverse branch in this section.

In Sect. 6.7, we have found a countable collection of disjoint open subsets $\sqcup _{i,k}\Delta _{p_i,k}$ in $\Delta (=\sqcup _i \Delta _{p_i})$ and an expanding map $T_0:\sqcup _{i,k}\Delta _{p_i,k}\rightarrow \Delta$. Without loss of generality, we may suppose that $T_0$ is irreducible, which means there doesn’t exist a nonempty subset of $I_1\subsetneq \{1,\ldots ,j \}$ such that

$$\begin{aligned} T_0(\cup _{i\in I_1}\Delta _{p_i})\subset \cup _{i\in I_1}\Delta _{p_i}. \end{aligned}$$

Otherwise, we can consider the restriction of $T_0$ to the union $\cup _{i\in I_1}\Delta _{p_i}$.

For $x\in \Delta _0=\Delta _{p_1}$, define the first return time

$$\begin{aligned} n(x)=\inf \{n\in {\mathbb {N}}:\,T_0^n(x)\in \Delta _{p_1} \}. \end{aligned}$$

Set $n(x)=\infty$ if $T_0^n(x)$ doesn’t come back to $\Delta _{p_1}$ for all $n\in {\mathbb {N}}$ or $T_0^n(x)$ lies outside of the domain of definition of $T_0$ for some n.

The expanding map T in Proposition 4.1 is defined by

$$\begin{aligned} T(x)=T_0^{n(x)}(x)\,\,\,\text {for}\,\,\, x\,\,\,\text {such that}\,\,\, n(x)<\infty . \end{aligned}$$

By the definition of $T_0$, we have

• either $T(x)=\gamma ^{-1}x$ with $\gamma \in {\mathcal {H}}_0$ and $\gamma : \Delta _{p_1}\rightarrow \Delta _{p_1}$,

• or $T(x)=\gamma _{n(x)}^{-1}\cdots \gamma _1^{-1} x$ with $\gamma _{l}\in {\mathcal {H}}_0$ for $l=1,\ldots ,n(x)$, where $\gamma _l$ maps $\Delta _{p_{k(l+1)}}$ to $\Delta _{p_{k(l)}}$ with $p_{k(1)}=p_{k(n(x)+1)}=p_1$ and $p_{k(l)}\ne p_1$ for $1<l\le n(x)$.

The string $\gamma _{n(x)}^{-1}\cdots \gamma _1^{-1}$ gives an open subset $\gamma _1\cdots \gamma _{n(x)}\Delta _{p_1}\subset \Delta _{p_1}$. They consist of open subsets described in Proposition 4.1.

To prove (1), (3) and (4) in Proposition 4.1, we start with a preliminary version of Lemma 4.8. Define

$$\begin{aligned} U_i=g_i^{-1}B(g_i\Delta _{p_i},1/(2\eta ))^c\,\,\,\text {for}\,\,\,1\le i\le j. \end{aligned}$$

Lemma 6.62

If $\gamma$ is an inverse branch in ${{\mathcal {H}}}_0$ which maps $\Delta _{p_i}$ into $\Delta _{p_l}$, then $\gamma ^{-1}U_l\subset U_i$.

Proof

Due to the construction, we know that $\gamma p_i$ is a parabolic fixed point inside $\Delta _{p_l}$. The definition of $U_l$ implies

$$\begin{aligned} g_lU_l\subset B(g_l\gamma p_i,1/(2\eta ))^c. \end{aligned}$$

Because the maps $g_i$’s are bi-Lipschitz (Lemma 5.12), we obtain

$$\begin{aligned} g_i U_l\subset B(g_i \gamma p_i,1)^c=B(g_i\gamma g_i^{-1}\infty ,1)^c. \end{aligned}$$

(6.63)

By Lemma 5.7, we have

$$\begin{aligned}&(g_i\gamma g_i^{-1})^{-1} B(g_i\gamma g_{i}^{-1}\infty , 1)^c=B((g_i\gamma g_i^{-1})^{-1}\infty , h(g_i \gamma p_i))\nonumber \\&\quad \subseteq B((g_i\gamma g_i^{-1})^{-1}\infty ,1). \end{aligned}$$

(6.64)

By (6.6), we obtain

$$\begin{aligned} d(g_i\Delta _{p_i}, (g_i\gamma g_i^{-1})^{-1}\infty )\ge 1/\eta . \end{aligned}$$

(6.65)

Combining (6.63)-(6.65) together, we conclude that

$$\begin{aligned} g_i \gamma ^{-1}U_l \subset B((g_i\gamma g_i^{-1})^{-1}\infty ,1)\subset B(g_i\Delta _{p_i},1/(2\eta ))^c. \end{aligned}$$

$\square$

We prove Proposition 4.1 (1) and (4.2). The proof is to consider an induced map and reduce the number of cusps by 1 at a time.

Let $q=p_j$. Denote $\cup _{1\le i\le j-1}\Delta _{p_i}=\Delta -\Delta _{q}$ by $X_1$ and for $x\in X_1$, define

$$\begin{aligned}n_1(x)=\inf \{n\in {\mathbb {N}}:\, T_0^n(x)\in X_1 \}.\end{aligned}$$

The map $T_1$ is given by $T_1(x)=T_0^{n_1(x)}(x)$ for x such that $n_1(x)<\infty$. Since $T_0$ is irreducible, this induced system is also irreducible on $X_1$. Write

$$\begin{aligned}&{\mathcal {H}}_q:=\text {the set of the inverse branches of}\,\,\,T_0\,\,\,\text {which are from}\,\,\,\Delta _q\,\,\,\text {to}\,\,\,\Delta _q,\\&{\mathcal {H}}_p:=\text {the set of the inverse branches of}\,\,\,T_0\,\,\,\text {which are from}\,\,\,X_1\,\,\,\text {to}\,\,\,X_1,\\&{\mathcal {H}}_{pq}:=\text {the set of the inverse branches of}\,\,\,T_0\,\,\,\text {which are from}\,\,\,X_1\,\,\,\text {to}\,\,\,\Delta _q,\\&{\mathcal {H}}_{qp}:=\text {the set of the inverse branches of}\,\,\,T_0\,\,\,\text {which are from}\,\,\,\Delta _q\,\,\,\text {to}\,\,\,X_1. \end{aligned}$$

As $T_1$ is a composition of multiples of $T_0$’s, we have

• either $T_1(x)=\gamma ^{-1}x$ with $\gamma \in {\mathcal {H}}_p$,

• or $T_1(x)=\gamma _{n_1(x)}^{-1}\cdots \gamma _1^{-1} x$ with $\gamma _1\in {\mathcal {H}}_{qp}$, $\gamma _{n_1(x)}\in {\mathcal {H}}_{pq}$ and $\gamma _l\in {\mathcal {H}}_q$ for $l=2,\ldots , n_1(x)-1$.

The string $\gamma _{1}\cdots \gamma _{n_1(x)}$ is an inverse branch of $T_1$. Set

$$\begin{aligned}&{{\mathcal {H}}}_1:=\text {the set of all inverse branches of}\,\,\, T_1,\\&{\mathcal {H}}_q^n:=\{\gamma _1\cdots \gamma _n:\,\gamma _i\in {\mathcal {H}}_q\,\,\,\text {for}\,\,\,1\le i\le n\}\,\,\,\text {for every}\,\,\,n\in {\mathbb {N}}. \end{aligned}$$

Lemma 6.66

There exists $C>0$ such that for every $n\in {\mathbb {N}}$ and for every $\gamma \in {{\mathcal {H}}}_q^n$, we have

$$\begin{aligned} |\gamma ' (x)|\ge |\gamma '|_\infty /C \,\,\,\text {for any}\,\,\,x\in \Delta _q. \end{aligned}$$

Proof

We first notice that $|\gamma '(x)|\approx |(g_j\gamma g_j^{-1})'(g_jx)|$. Write $p=\gamma p_j$. By Lemma 5.7, we have

$$\begin{aligned}| (g_j\gamma g_j^{-1})' (y)|=\frac{h(g_j p)}{d(y,g_j\gamma ^{-1}p_j)^2}.\end{aligned}$$

By (6.6), we have $d(g_j\Delta _q,g_j\gamma ^{-1}p_j)=d(g_j\Delta _q,(g_j\gamma g_j^{-1})^{-1}\infty )>1/(2\eta )$. Then for every $y\in g_j\Delta _{p_j}=g_j\Delta _q$, the distance $d(y,g_j\gamma ^{-1}p_j)\in [d(g_j\Delta _q,g_j\gamma ^{-1}p_j)\pm {\text {diam}}(g_j\Delta _q)]$, which implies the lemma. $\square$

Lemma 6.67

There exist $C>0$, $\epsilon >0$ such that for every $l\in {\mathbb {N}}$

$$\begin{aligned} \sum _{1\le k\le l,\ \gamma _k\in {{\mathcal {H}}}_q}|(\gamma _1\cdots \gamma _l)'|_\infty ^{\delta }<C(1-\epsilon )^l. \end{aligned}$$

Proof

Claim: there exists $\epsilon >0$ such that for every $n\in {\mathbb {N}}$ and for any $h\in {{\mathcal {H}}}_q^n$, we have

$$\begin{aligned} \sum _{\gamma \in {{\mathcal {H}}}_q}\mu (h\gamma \Delta _q)\le (1-\epsilon )\mu (h\Delta _q). \end{aligned}$$

(6.68)

Proof of the claim: for a measurable set $E\subset \Delta _q$, by Lemma 6.66

$$\begin{aligned} \mu (hE)=\int _E |h'(x)|_{{\mathbb {S}}^d}^\delta \mathrm {d}\mu (x)\approx \int _E |h'(x)|^\delta \mathrm {d}\mu (x) \in \mu (E) |h'|^{\delta }_\infty [1/C,1].\nonumber \\ \end{aligned}$$

(6.69)

Write $F=\cup _{\gamma \in {{\mathcal {H}}}_q} \gamma \Delta _q$. Since $T_0$ is irreducible, we have $\mu (F)<\mu (\Delta _q)$. By (6.69)

$$\begin{aligned}&\sum _{\gamma \in {{\mathcal {H}}}_q}\mu (h\gamma \Delta _q)=\mu (hF)\le |h'|^\delta _\infty \mu (F)\\&\quad =\frac{\mu (F)}{\mu (\Delta _q-F)}|h'|^\delta _\infty \mu (\Delta _q-F)\le C' \mu (h(\Delta _q-F)). \end{aligned}$$

So we have

$$\begin{aligned} (1+1/C')\sum _{\gamma \in {\mathcal {H}}_q}\mu (h\gamma \Delta _q)\le \mu (hF)+\mu (h(\Delta _q-F))\le \mu (h\Delta _q). \end{aligned}$$

Using (6.68), Lemma 6.66 and (6.69) with $E=\Delta _q$, we obtain

$$\begin{aligned}&\sum _{1\le k\le l,\ \gamma _k\in {{\mathcal {H}}}_q}|(\gamma _1\cdots \gamma _l)'|_\infty ^{\delta }\le C \sum _{1\le k\le l,\ \gamma _k\in {{\mathcal {H}}}_q}\mu (\gamma _1\cdots \gamma _l\Delta _q)\\&\qquad \le C \sum _{1\le k\le l-1,\ \gamma _k\in {{\mathcal {H}}}_q}(1-\epsilon )\mu (\gamma _1\cdots \gamma _{l-1}\Delta _q) \le C (1-\epsilon )^l. \end{aligned}$$

$\square$

Proof of Proposition 4.1 (1) and (4.2)

We first use Proposition 6.15, (6.58), Lemma 6.66 and 6.67 to prove that for the expanding map $T_1$, we have

• 1. There exists $\epsilon _1>0$ such that

  $$\begin{aligned} \sum _{\gamma \in {{\mathcal {H}}}_1}|\gamma '|_\infty ^{\delta -\epsilon _1}<\infty , \end{aligned}$$

  (6.70)

  where $|\gamma '|_\infty$ is defined as in (6.56).

• 2. There exists $\epsilon >0$ such that for every $n\in {\mathbb {N}}$,

  $$\begin{aligned} \mu (\{x\in X_1:\,n_1(x)>n+1\})\ll (1-\epsilon )^n. \end{aligned}$$

  (6.71)

The second statement in particular implies that the map $T_1$ is defined almost everywhere in $\Delta -\Delta _q$.

Due to Lemma 6.67, we can find a large $l_0$ such that $\sum _{1\le k\le l_0,\ \gamma _k\in {{\mathcal {H}}}_q}|(\gamma _1\cdots \gamma _{l_0})'|_\infty ^{\delta }<1$. Then using (6.58) and submultiplicativity $|(\gamma _1\gamma _2)'|_\infty \le |(\gamma _1)'|_\infty |(\gamma _2)'|_\infty$, we obtain

$$\begin{aligned} \sum _{1\le k\le l_0,\ \gamma _k\in {{\mathcal {H}}}_q}|(\gamma _1\cdots \gamma _{l_0})'|_\infty ^{\delta -\epsilon }<\infty , \end{aligned}$$

where $\epsilon >0$ is the constant given by (6.58). Hence we can find $0<\epsilon _1<\epsilon$ small such that

$$\begin{aligned} \sum _{1\le k\le l_0,\ \gamma _k\in {{\mathcal {H}}}_q}|(\gamma _1\cdots \gamma _{l_0})'|_\infty ^{\delta -\epsilon _1}<1. \end{aligned}$$

Using submultiplicativity, we obtain constants $C>0,\rho <1$ such that for $l\in {\mathbb {N}}$

$$\begin{aligned} \sum _{1\le k\le l,\ \gamma _k\in {{\mathcal {H}}}_q}|(\gamma _1\cdots \gamma _{l})'|_\infty ^{\delta -\epsilon _1}\le C\rho ^l. \end{aligned}$$

(6.72)

Denote $\sum _{\gamma \in {\mathcal {H}}_1}|\gamma '|_{\infty }^{\delta -\epsilon _1}$ by $E_q$. For every inverse branch of $T_1$, it can be uniquely decomposed as $\gamma _0\gamma _1\cdots \gamma _{l}\gamma _{l+1}$ with $\gamma _{l+1}\in {{\mathcal {H}}}_{pq}$, $\gamma _{i}\in {{\mathcal {H}}}_q$ with $i=1,\cdots l$ and $\gamma _0\in {{\mathcal {H}}}_{qp}$. Using this expression and submultiplicativity, we obtain

$$\begin{aligned}&E_q\le \sum _{\gamma \in {{\mathcal {H}}}_p}|\gamma ' |_\infty ^{\delta -\epsilon _1}+\sum _{l\ge 1} \left( \sum _{\gamma _0\in {{\mathcal {H}}}_{qp}} |\gamma _0'|_\infty ^{\delta -\epsilon _1}\right) \left( \sum _{\gamma _{l+1}\in {{\mathcal {H}}}_{pq}} |\gamma _{l+1}'|_\infty ^{\delta -\epsilon _1}\right) \nonumber \\&\quad \left( \sum _{\gamma _i\in {{\mathcal {H}}}_q,1\le i\le l}|(\gamma _1\cdots \gamma _l)'|_\infty ^{\delta -\epsilon _1}\right) . \end{aligned}$$

(6.73)

Therefore $E_p$ is also finite due to (6.73), (6.72) and (6.58).

The set of x such that $T_0^n(x)$ is outside of domain of definition of $T_0$ for some n has zero PS measure by Proposition 6.15. We only need to consider the set of x such that $T_0^n(x)$ is in the domain of definition for every n. If $x\in X_1$ with $n_1(x)>n+1$, then x must be in $\gamma _0\gamma _1\cdots \gamma _n\Delta _q$ with $\gamma _0\in {{\mathcal {H}}}_{qp}$ and $\gamma _i\in {{\mathcal {H}}}_q$ for $1\le i\le n$. Therefore Lemma 6.67 implies

$$\begin{aligned}&\mu (\{x\in X_1:\,n_1(x)>n+1\})\le \sum _{\gamma _0\in {{\mathcal {H}}}_{qp}}\sum _{\gamma _i\in {{\mathcal {H}}}_q,1\le i\le n}\mu ( \gamma _0\gamma _1\cdots \gamma _n\Delta _q)\\&\qquad \le \left( \sum _{\gamma _0\in {{\mathcal {H}}}_{qp}} |\gamma _0'|_\infty ^{\delta }\right) \left( \sum _{\gamma _i\in {{\mathcal {H}}}_q,1\le i\le n}|(\gamma _1\cdots \gamma _n)'|_\infty ^{\delta }\right) \ll (1-\epsilon )^n. \end{aligned}$$

We keep reducing the number of cusps by considering the set $X_2:=X_1-\Delta _{p_{j-1}}$ and the induced map $T_2:X_2\rightarrow X_2$ which is constructed similar to $T_1$: in particular, the inverse branches of $T_2$ are compositions of elements in ${\mathcal {H}}_1$. Analogs of Lemma 6.66 and 6.67 for $T_2$ also hold. The replacements of Proposition 6.15 and (6.58) are (6.71) and (6.70) respectively. Using these three ingredients, we can show the properties like (6.70) and (6.71) also hold for $T_2$. The proof of Proposition 4.1(1) and (4.2) will be finished by repeating this. $\square$

Now, we will finish proving the rest of results for the coding except Lemma 4.5 (UNI).

Proof of Lemma 4.8

Take

$$\begin{aligned} \Lambda _{-}=\Lambda _\Gamma \cap \{|x|>1/(2\eta ) \}=\Lambda _\Gamma \cap U_1. \end{aligned}$$

(6.74)

The contracting map $\gamma$ from $\Delta _0$ to $\Delta _0$ is a composition of maps in ${{\mathcal {H}}}_0$, so the inclusion follows directly from Lemma 6.62. Write $p=\gamma \infty$. By Lemma 5.7,

$$\begin{aligned} |(\gamma ^{-1})'(x)|_{{\mathbb {S}}^d}=\frac{h(p)}{d(x,p)^2}\frac{1+|x|^2}{1+|\gamma ^{-1}x|^2}. \end{aligned}$$

For $x\in \Lambda _{-}$, as $p=\gamma \infty \in \Delta _0$, we have

$$\begin{aligned} \frac{1+|x|^2}{d(x,p)^2}\le \frac{1+|x|^2}{(|x|-{\text {diam}}(\Delta _0))^2}. \end{aligned}$$

The right hand side of the inequality is around 1 as $|x|\ge 1/(2\eta )$. For $\gamma ^{-1}x\in \Lambda _{ -}$, we have $|\gamma ^{-1}x|\ge 1/(2\eta )$. Hence $|(\gamma ^{-1})'(x)|_{{\mathbb {S}}^d}\le \lambda$ for some $\lambda$ independent of $\gamma$. $\square$

Proof of Proposition 4.1 (3)

By Lemma 5.7, we have

$$\begin{aligned} |\gamma '(x)|=\frac{h(p)}{d(x,\gamma ^{-1}\infty )^2}. \end{aligned}$$

By Lemma 4.8, we have $\gamma ^{-1}\infty \in \Lambda _{-}$, which implies $d(\gamma ^{-1}\infty ,x)\ge 1/(2\eta )$. Hence

$$\begin{aligned} |\gamma '(x)|\le (2\eta )^2h(p)\le 4\eta ^2.\end{aligned}$$

$\square$

Proof of Proposition 4.1(4). We need the following lemma, which will also be needed in later sections.

Lemma 6.75

Let $\gamma$ be any element in $\Gamma$ which does not fix $\infty$. For any $x\in \Delta _0$ and any unit vector $e\in {\mathbb {R}}^d$, we have

$$\begin{aligned}\partial _e\log | \gamma '(x)|=-\frac{2\langle x-\xi ,e \rangle }{|x-\xi |^2} \end{aligned}$$

where $\xi =\gamma ^{-1}\infty$.

Proof

It can be shown using Lemma 5.7 and elementary computation. $\square$

For $\gamma \in {{\mathcal {H}}}$, Proposition 4.1 (4) can be deduced using Lemma 6.75 and the observation that $|\gamma ^{-1}\infty |\ge 1/(2\eta )$ (Lemma 4.8 and (6.74)).

6.9 Verifying UNI

We prove Lemma 4.5 in this part. Let $\Gamma _f$ be the semigroup generated by $\gamma$ in ${{\mathcal {H}}}$ and $\Gamma _b$ be the semigroup generated by $\gamma ^{-1}$ with $\gamma \in {{\mathcal {H}}}$. Let $\Lambda _f$ and $\Lambda _b$ be the limit set of $\Gamma _f$ and $\Gamma _b$ on $\partial {\mathbb {H}}^{d+1}$, that is the set of accumulation points of orbit $\Gamma _f o$ and $\Gamma _b o$ for some $o\in {\mathbb {H}}^{d+1}$ respectively. It follows from the definition that the limit set $\Lambda _f$ is $\Gamma _f$-invariant and $\Lambda _b$ is $\Gamma _b$-invariant. Due to [24, Proposition 3.19] (convergence property of Möbius transformation), we have that $\Lambda _f$ is a $\Gamma _f$-minimal set and $\Lambda _{b}$ is a $\Gamma _b$-minimal set.

Lemma 6.76

The limit set $\Lambda _b$ is not contained in an affine subspace in ${\mathbb {R}}^d\cup \{\infty \}$ or a sphere in ${\mathbb {R}}^d$.

Proof

Let A be an affine subspace or a sphere with minimal dimension which contains $\Lambda _b$. Because $\Lambda _b$ is $\Gamma _b$ invariant, the semigroup $\Gamma _b$ must preserve A, so does the Zariski closure of $\Gamma _b$. The Zariski closure of a semigroup is a group (see for example [13, Lemma 6.15]). The Zariski closures of $\Gamma _f$ and $\Gamma _b$ are the same. Hence $\Gamma _f$ also preserves A and $\Lambda _f$ is in A. We claim: $\mu (\Lambda _f)=\mu (\Lambda _{\Gamma }\cap \Delta _0)>0$. Then because $\Gamma$ is Zariski dense, by [20, Corollary 1.4], we conclude that $\mu (A)$ is non zero if and only if $A={\mathbb {R}}^d$, finishing the proof.

Proof of the claim: Let x be any point in $\Lambda _{\Gamma }\cap \Delta _0$ such that $T^nx\in \Lambda _{\Gamma }\cap \Delta _0$ for every $n\in {\mathbb {N}}$. We can write $x=\gamma _n T^n(x)\in \gamma _n\Delta _0$ for some $\gamma _n\in {{\mathcal {H}}}^n$. Fix any $y\in \Lambda _f$, it follows from Proposition 4.1 (3) that $d(\gamma _n y, \gamma _n T^n(x))\rightarrow 0$. So $\gamma _n y\rightarrow x$ and $x\in \Lambda _f$. Due to Proposition 4.1 (1), the set of x’s such that $T^nx\in \Lambda _{\Gamma }\cap \Delta _0$ for every $n\in {\mathbb {N}}$ is a conull set in $\Lambda _{\Gamma }\cap \Delta _0$. Hence $\mu (\Lambda _f)=\mu (\Lambda _{\Gamma }\cap \Delta _0)$. $\square$

Lemma 6.77

For every $x\in \Lambda _\Gamma \cap {{\overline{\Delta }}}_0$, there exist pairs of points $(\xi _{1m},\xi _{2m}),\ m=1,\ldots , k_x$ in the limit set $\Lambda _b$ and $\epsilon _x'>0$ such that for every unit vector $e\in {\mathbb {R}}^d$ there exists m,

$$\begin{aligned} \left| \left\langle \frac{x-\xi _{1m}}{|x-\xi _{1m}|^2}-\frac{x-\xi _{2m}}{|x-\xi _{2m}|^2},e \right\rangle \right|>2\epsilon _x'>0. \end{aligned}$$

Proof

The map $inv_x:\xi \mapsto \frac{x-\xi }{|x-\xi |^2}$ is an inversion and this map is injective. If there exists a unit vector $e\in {\mathbb {R}}^d$ such that

$$\begin{aligned} \left\langle \frac{x-\xi _{1}}{|x-\xi _{1}|^2}-\frac{x-\xi _{2}}{|x-\xi _{2}|^2},e \right\rangle =0 \end{aligned}$$

for all $\xi _1,\xi _2$ in $\Lambda _b$, then $inv_x(\Lambda _b)$ is contained in an affine subspace parallel to $e^\perp$. Hence $\Lambda _b$ itself is contained in an affine subspace in ${\mathbb {R}}^d\cup \{\infty \}$ or a sphere in ${\mathbb {R}}^d$, which contradicts Lemma 6.76. Therefore, for every unit vector $e\in {\mathbb {R}}^d$, there exist $\xi _1,\xi _2$ in $\Lambda _b$ such that

$$\begin{aligned}\left\langle \frac{x-\xi _{1}}{|x-\xi _{1}|^2}-\frac{x-\xi _{2}}{|x-\xi _{2}|^2},e \right\rangle \ne 0. \end{aligned}$$

We use continuity and compactness to finish the proof. $\square$

Lemma 6.78

Let $\xi$ be any point in $\Lambda _b$. For any $\epsilon _2,\epsilon _3>0$, there exists $n_\xi \in {\mathbb {N}}$ such that for any $n\ge n_\xi$, there exists $\gamma$ in ${{\mathcal {H}}}^n$ satisfying

$$\begin{aligned} d_{{\mathbb {S}}^d}(\gamma ^{-1}\infty ,\xi )\le \epsilon _2,\ |\gamma '|_\infty \le \epsilon _3. \end{aligned}$$

Proof

Since $\Lambda _b$ is $\Gamma _b$ minimal, for any point $\xi '\in \Lambda _b$, there exists a sequence $\{\gamma _n^{-1}\}$ in $\Gamma _b$ such that $\gamma _n^{-1}\xi '$ converges to $\xi$ and $|\gamma _n'|_\infty$ tends to zero. By Lemma 4.8, we know that $\gamma _n^{-1}\Lambda _{-}$ also converges to $\xi$. Hence we can always find a $\gamma$ in $\Gamma _f$ with $|\gamma '|_\infty \le \epsilon _3$ and $d_{{\mathbb {S}}^d}(\gamma ^{-1}\Lambda _{-},\xi )\le \epsilon _2$. Let $n_\xi$ be the unique number such that $\gamma \in {{\mathcal {H}}}^{n_\xi }$.

For any $\gamma _1\in \cup _{n\ge 1}{{\mathcal {H}}}^{n}$, we have $|(\gamma _1\gamma )'|_\infty \le |\gamma _1'|_\infty |\gamma '|_\infty \le |\gamma '|_\infty$ and

$$\begin{aligned}d_{{\mathbb {S}}^d}((\gamma _1\gamma )^{-1}\infty ,\xi )=d_{{\mathbb {S}}^d}(\gamma ^{-1}(\gamma _1^{-1}\infty ),\xi )\le d_{{\mathbb {S}}^d}(\gamma ^{-1}\Lambda _{-},\xi )\le \epsilon _2. \end{aligned}$$

Therefore, for any $m= n_\xi +n$, choose any $\gamma _1\in {{\mathcal {H}}}^n$ and then $\gamma _1\gamma \in {{\mathcal {H}}}^m$ and it satisfies Lemma 6.78. $\square$

Combining the above two lemmas, by Lemma 6.75 and the formula $R_n(\gamma x)=-\log |\gamma '(x)|$ for $\gamma \in {{\mathcal {H}}}^n$, we have

$$\begin{aligned}|\partial _e(R_n\circ \gamma _{1m}-R_n\circ \gamma _{2m})(y)|=\left| \left\langle \frac{y-\gamma _{1m}^{-1}\infty }{|y-\gamma _{1m}^{-1}\infty |^2}-\frac{y-\gamma _{2m}^{-1}\infty }{|y-\gamma _{2m}^{-1}\infty |^2},e \right\rangle \right| . \end{aligned}$$

Using this expression, Lemma 6.77, 6.78 and continuity, we obtain

Lemma 6.79

For every $x\in \Lambda _\Gamma \cap {{\overline{\Delta }}}_0$, there exist $\epsilon _x, \epsilon _x'>0$ such that for any $\epsilon _3>0$, there exists $n_x\in {\mathbb {N}}$ such that the following holds for any $n\ge n_x$. There exist $k_x\in {\mathbb {N}}$, $\gamma _{im}\in {{\mathcal {H}}}^n$ with $i=1,2$ and $m=1,\ldots , k_x$ satisfying

• $|\gamma '_{im}|<\epsilon _3$ for every $i=1,2$ and $m=1,\ldots ,k_x$.

• for any unit vector $e\in {\mathbb {R}}^d$, there exists $m\in \{1,\ldots ,k_x\}$ such that for any $y\in B(x,\epsilon _x)$,

  $$\begin{aligned}|\partial _e(R_n\circ \gamma _{1m}-R_n\circ \gamma _{2m})(y)|\ge \epsilon _x'>0. \end{aligned}$$

Proof of Lemma 4.5

For every $x\in \Lambda _\Gamma \cap {{\overline{\Delta }}}_0$, we apply Lemma 6.79 to x and get two constants $\epsilon _x, \epsilon _x'>0$. Since $\Lambda _\Gamma \cap {{\overline{\Delta }}}_0$ is compact, we can find a finite set $\{x_1,\ldots ,x_l \}$ such that $\cup B(x_j,\epsilon _{x_j}/2)\supset \Lambda _\Gamma \cap {{\overline{\Delta }}}_0$. Let $\epsilon _0=\inf \{\epsilon _{x_j}' \}$ and $r=\inf \{\epsilon _{x_j}/2 \}$. Take $\epsilon _3=\epsilon _0/C$ and $n_0\ge \sup _{1\le j\le l}\{n_{x_j} \}$. Then for every $x_j$, there exists a finite set $\{\gamma _{im}\}$ in ${\mathcal {H}}^{n_0}$ satisfying results in Lemma 6.79. We put all these $\gamma _{im}$’s together and this is the finite set in ${\mathcal {H}}^{n_0}$ described in Lemma 4.5. For any $x\in \Lambda _\Gamma \cap {{\overline{\Delta }}}_0$, it is contained in some $B(x_j,\epsilon _{x_j}/2)$. Then $B(x,r)\subset B(x_j,\epsilon _{x_j})$. The family $\{\gamma _{im}\}$ for $x_j$ will satisfy nonvanishing condition on $B(x,r)$, that is for every unit vector $e\in {\mathbb {R}}^d$ there exists m such that for any $y\in B(x,r)$

$$\begin{aligned}|\partial _e(R_{n_0}\circ \gamma _{1m}-R_{n_0}\circ \gamma _{2m})(y)|\ge \epsilon _0>0. \end{aligned}$$

Finally, the inequality $|\mathrm D\tau _{im}|_\infty \le C_2$ is due to (4.4).

$\square$

7 Spectral gap and Dolgopyat-type spectral estimate

In this section, we prove a Dolgopyat-type spectral estimate and the main result is Proposition 7.3. Our argument is influenced by the one in [1, 2, 5, 18, 34, 47] and there is some technical variation in the current setting. The proof involves proving a cancellation lemma (Lemma 7.14) and using it to obtain $L^2$ contraction. The rough idea is as follows. Denote the set $\Delta _0\cap \Lambda _{\Gamma }$ by $\Lambda _0$. With the UNI property (Lemma 4.5) available, for each ball B(y, r) with $y\in \Lambda _0$, one uses the doubling property of the PS measure to find a point $x\in B(y,r)\cap \Lambda _0$ such that cancellation happens on $B(x,r')$. Then, to run the classical argument, one needs to find finitely many such pairwise disjoint balls $B(x_i,r')$’s contained in $\Delta _0$ such that $\sqcup B(x_i,Dr')$ covers $\Delta _0$ for some $D>1$. The difficulty lies in that the balls $B(x_i,r)$’s are produced using PS measure so the position of $B(x_i,r')$’s is in some sense random and some $B(x_i,r')$ may not be fully contained in $\Delta _0$. To overcome this, we find $B(x_i,r')$’s which only cover a subset of $\Delta _0$ and divide the proof of Proposition 7.3 into the cases when the iteration is small and when the iteration is large.

7.1 Twisted transfer operators

For $s\in {\mathbb {C}}$, let $L_s$ be the twisted transfer operator defined by

$$\begin{aligned} L_s(u)(x)=\sum _{\gamma \in {{\mathcal {H}}}} |\gamma '(x)|^{\delta +s}u(\gamma x). \end{aligned}$$

(7.1)

For $u:\Delta _0\rightarrow {\mathbb {C}}$, define

$$\begin{aligned} \Vert u\Vert _{\text {Lip}}=\max \{|u|_{\infty },\,|u|_{\text {Lip}}\}, \end{aligned}$$

where $|u|_{\text {Lip}}=\sup _{x\ne y} |u(x)-u(y)|/d(x,y)$, where $d(\cdot , \cdot )$ is the Euclidean distance. Denote by $\text {Lip}(\Delta _0)$ the space of functions $u:\Delta _0\rightarrow {\mathbb {C}}$ with $\Vert u\Vert _{\text {Lip}}<\infty$. We also introduce a family of equivalent norms on $\text {Lip}(\Delta _0)$:

$$\begin{aligned} \Vert u\Vert _b=\max \{|u|_{\infty },\,|u|_{\text {Lip}}/(1+|b|)\},\,\,\,b\in {\mathbb {R}}. \end{aligned}$$

With Proposition 4.1 available, we obtain the following lemma by a verbatim of the proof of [1, Proposition 2.5].

Lemma 7.2

Write $s=\sigma +i b$. The family $s\mapsto L_s$ of operators on ${\text {Lip}}(\Delta _0)$ is continuous on $\{s\in {\mathbb {C}}:\ \sigma >-\epsilon _o\}$, where $\epsilon _o$ is given as in Proposition 4.1 (4). Moreover, $\sup _{|\sigma |<\epsilon _o} \Vert L_s\Vert _b<\infty$.

Define the PS measure $\mu _E$ on $\Delta _0$ with respect to the Euclidean metric by

$$\begin{aligned}\mathrm {d}\mu _E(x)=(1+|x|^2)^\delta \mathrm {d}\mu (x). \end{aligned}$$

Using the quasi-invariance of the PS measure $\mu$, we obtain that the dual operator of $L_0$ preserves the measure $\mu _E$ by a straightforward computation. Our main result of Dolgopyat argument is the following $L^2$ contracting proposition.

Proposition 7.3

There exist $C>0,\ \beta <1, \epsilon >0$ and $b_0>0$ such that for all v in ${\text {Lip}}(\Delta _0)$, $m\in {\mathbb {N}}$ and $s=\sigma +ib$ with $|\sigma |\le \epsilon$ and $|b|>b_0$, we have

$$\begin{aligned} \int |L_s^{m}v|^2\mathrm {d}\mu _E\le C\beta ^m\Vert v\Vert ^2_b. \end{aligned}$$

The proof will be given at the end of Sect. 7.5.

Recall that $\nu$ is the unique T-invariant ergodic probability measure on $\Delta _0\cap \Lambda _{\Gamma }$ which is absolutely continuous with respect to the PS measure $\mu$ with a positive Lipschitz density function ${{\bar{f}}}_0$. So $\nu$ is also absolutely continuous with respect to $\mu _{E}$ with a positive Lipschitz density function $f_0$. Based on these, it is a classical result that the operator $L_0$ acts on ${\text {Lip}}(\Delta _0)$ and has a spectral gap and a simple isolated eigenvalue at 1 with $f_0$ the corresponding eigenfunction.

For $\sigma \in {\mathbb {R}}$ close enough to 0, $L_{\sigma }$ acting on $\text {Lip}(\Delta _0)$ is a continuous perturbation of $L_0$ (see Lemma 7.2). Hence, it has a unique eigenvalue $\lambda _{\sigma }$ close to 1, and the corresponding eigenfunction $f_{\sigma }$ (normalized so that $\int f_{\sigma }=1$) belongs to $\text {Lip}(\Delta _0)$, strictly positive, and tends to $f_0$ in $\text {Lip}(\Delta _0)$ as $\sigma \rightarrow 0$. Choose a sufficiently small $\epsilon \in (0,\epsilon _o)$ such that for $\sigma \in (-\epsilon ,\epsilon )$, $f_{\sigma }$ is well defined and

$$\begin{aligned} 1/2\le \lambda _{\sigma }\le 2,\,\,\, f_{0}/2\le f_{\sigma }\le 2 f_{0},\,\,\, |f_0|_{\text {Lip}}/2\le |f_{\sigma }|_{\text {Lip}}\le 2|f_0|_{\text {Lip}}. \end{aligned}$$

For $s=\sigma +ib$ with $|\sigma |<\epsilon$ and $b\in {\mathbb {R}}$, define a modified transfer operator ${\tilde{L}}_{s}$ by

$$\begin{aligned} {\tilde{L}}_{s}(u)=(\lambda _{\sigma }f_{\sigma })^{-1}L_{s}(f_{\sigma }u). \end{aligned}$$

(7.4)

It satisfies ${\tilde{L}}_{\sigma }1=1$, and $|{\tilde{L}}_{s}u|\le {\tilde{L}}_{\sigma }|u|$.

Lemma 7.5

(Lasota-Yorke inequality) There is a constant $C_{16}>1$ such that

$$\begin{aligned} |{\tilde{L}}^n_s v|_{{\text {Lip}}} \le C_{16} (1+|b|) |v|_{\infty } +C_{16} \lambda ^n |v|_{{\text {Lip}}} \end{aligned}$$

(7.6)

holds for any $s=\sigma +ib$ with $|\sigma |<\epsilon$, and all $n\ge 1,\,v\in {\text {Lip}}(\Delta _0)$, where $\lambda$ is given as in Proposition 4.1.

The proof of this lemma is a verbatim of proof of [1, Lemma 2.7]. The following lemma can be deduced from Lemma 7.5 by a straightforward computation.

Lemma 7.7

We have $\Vert {\tilde{L}}^n_s\Vert _b\le 2C_{16}$ for all $s=\sigma +ib$ with $|\sigma |<\epsilon$ and all $n\ge 1$.

7.2 Cancellation lemma

The main result of this subsection is the cancellation lemma (Lemma 7.14) and the proof is inspired by the proof of analogous results in [34] and [47]. We start with detailing all the constants.

Let $C_{17}$ be the constant which will be specified in (7.28). We define the cone

Definition 7.8

For $b\in {\mathbb {R}}$, let

$$\begin{aligned} {\mathcal {C}}_b=&\{(u,v):\,u,\,v\in \text {Lip}(\Delta _0),\,u>0,\,0\le |v|\le u,\,|\log u|_{\text {Lip}}\le C_{17} |b|,\\&|v(x)-v(y)|\le C_{17} |b| u(y)d(x,y)\,\,\,\text {for all}\,\,\,x,y\in \Delta _0\}. \end{aligned}$$

Let $r>0$ and $\epsilon _0>0$ be the same constants as the ones in Lemma 4.5. We apply Lemma 4.5 with $C=16C_{17}$. Let $n_0$ be a sufficiently large integer which satisfies Lemma 4.5 and the inequality

$$\begin{aligned} \lambda ^{n_0}C_{17}(1+{\text {diam}}(\Delta _0))\le 1. \end{aligned}$$

(7.9)

Let $\gamma _{mj}$, with $m=1,2$, $j=1,\ldots ,j_0$ be the inverse branches given by Lemma 4.5.

Let $k\in {\mathbb {N}}$ be such that

$$\begin{aligned} k\epsilon _0>16(C_2+\epsilon _0), \end{aligned}$$

(7.10)

where $C_2$ is given in (4.3).

Note that the measure $\nu$ is absolutely continuous with respect to the PS measure $\mu$. Let $D>0$ be such that for all $x\in \Lambda _\Gamma \cap \Delta _0$ and $r'\le 1/C_{3}$ with $C_{3}$ given in Proposition 5.14

$$\begin{aligned} \nu (B(x,Dr'))>\nu (B(x,(k+2)r')). \end{aligned}$$

(7.11)

Let $\epsilon _2>0$ be such that

$$\begin{aligned} (2C_{17}\epsilon _2+1/4)e^{2C_{17}\epsilon _2}\le 3/4. \end{aligned}$$

(7.12)

Let $\epsilon _3>0$ be such that

$$\begin{aligned} \epsilon _3(D+2)< \min \{\epsilon _2,\ r,\ 1/C_{3}\},\ \epsilon _3(D+2)(C_2+\epsilon _0)<3\pi /2,\ \epsilon _3k \epsilon _0<\pi . \end{aligned}$$

(7.13)

Recall the notation $\tau _{mj}$ introduced in Lemma 4.5. For $s=\sigma +ib\in {\mathbb {C}}$, define

$$\begin{aligned} A_{s,\gamma _{mj}}(v)(x)=e^{(s+\delta )\tau _{mj}(x)}f_\sigma (\gamma _{mj}x)v(\gamma _{mj}x). \end{aligned}$$

Lemma 7.14

There exists $0< \eta _0<1$ such that the following holds. For $s=\sigma +ib$ with $|\sigma |\le \epsilon$, $|b|>1$, for $(u,v)\in C_b$, and for any $y\in \Lambda _0$, there exists $x\in B(y,\epsilon _3D/|b|)\cap \Lambda _\Gamma$ such that we have the following: there exists $j\in \{1,\ldots ,j_0\}$ such that one of the following inequalities holds on $B(x,\epsilon _3/|b|)$:

$$\begin{aligned} {\textbf {type }} \gamma _{1j}:\ |A_{s,\gamma _{1j}}(v)+A_{s,\gamma _{2j}}(v)|\le \eta _0A_{\sigma ,\gamma _{1j}}(u)+A_{\sigma ,\gamma _{2j}}(u),\\ {\textbf {type }} \gamma _{2j}:\ |A_{s,\gamma _{1j}}(v)+A_{s,\gamma _{2j}}(v)|\le A_{\sigma ,\gamma _{1j}}(u)+\eta _0 A_{\sigma ,\gamma _{2j}}(u). \end{aligned}$$

We first prove a quick estimate.

Lemma 7.15

Let $\epsilon _2$ be the constant defined in (7.12). For any $|b|>1$, for $(u,v)\in C_b$ and for a ball Z of radius $\epsilon _2/|b|$, we have

• 1. $\inf _Zu\ge e^{-2C_{17}\epsilon _2}\sup _Zu;$

• 2. either $|v|\le \frac{3}{4}u$ for all $x\in Z$ or $|v|\ge \frac{1}{4}u$ for all $x\in Z$.

Proof

The first inequality is due to $|\log u(x)-\log u(y)| \le C_{17}|b||x-y|$ for every $x,y\in \Delta _0$.

Suppose there exists $x_0\in Z$ such that $|v(x_0)|\le \frac{1}{4}u(x_0)$. Then by (7.12)

$$\begin{aligned} |v(x)|&\le |v(x)-v(x_0)|+\frac{1}{4}u(x_0)\le C_{17}|x-x_0||b|\sup _Zu+\frac{1}{4}\sup _Zu\\&\le \left( 2C_{17}\epsilon _2+\frac{1}{4}\right) \sup _Zu\le (2C_{17}\epsilon _2+\frac{1}{4})e^{2C_{17}\epsilon _2}\inf _Z u\le \frac{3}{4}u(x). \end{aligned}$$

$\square$

Proof of Lemma 7.14

It follows (7.11) that there exists $x_0\in (B(y,\epsilon _3 D/|b|)-B(y,(k+2)\epsilon _3/|b|))\cap \Lambda _\Gamma$. Let $B_1=B(y,\epsilon _3/|b|)$, $B_2=B(x_0,\epsilon _3/|b|)$ and ${\hat{B}}$ the smallest ball containing $B_1\cup B_2$. For all $x\in B_1,x'\in B_2$, we have

$$\begin{aligned} d(x,x')\in \frac{\epsilon _3}{b}[k,D+2]. \end{aligned}$$

(7.16)

In view of (7.13), the radius of ${\hat{B}}$ is smaller than $\epsilon _2/|b|$ and it is contained in B(y, r). Let $e_0=(y-x_0)/|y-x_0|$.

By Lemma 4.5 for the point y there exists j in $\{1,\ldots ,j_0\}$ such that (4.6) holds for B(y, r) with $e=e_0$. From now on, j is fixed, so we abbreviate $(\gamma _{1j},\gamma _{2j})$ to $(\gamma _1,\gamma _2)$ and $(\tau _{1j},\tau _{2j})$ to $(\tau _1,\tau _2)$.

Due to $|\gamma _m'|_\infty \le \lambda \le 1$, the radius of $\gamma _{m}{\hat{B}}$ is smaller than $\epsilon _{2}/|b|$. So we can apply Lemma 7.15 to $\gamma _{m}{\hat{B}}$ and we have that either $|v(\gamma _mx)|\ge \frac{1}{4} u(\gamma _mx)$ for all $x\in {\hat{B}}$ or $|v(\gamma _mx)|\le \frac{3}{4} u(\gamma _mx)$ for all $x\in {\hat{B}}$. Suppose that

$$\begin{aligned}|v(\gamma _mx)|\le \frac{3}{4}u(\gamma _mx) \end{aligned}$$

holds for some $m\in \{1,2 \}$ for all $x\in {{\hat{B}}}$. Then Lemma 7.14 can be proved by a straightforward computation.

Suppose that for $x\in {\hat{B}}$ and $m=1,2$

$$\begin{aligned} |v(\gamma _mx)|\ge \frac{1}{4}u(\gamma _mx). \end{aligned}$$

(7.17)

Claim

Under the assumption of (7.17), there exists $C_{18}>0$ independent of b and (u, v) such that for $l\in \{1,2\}$, we have

$$\begin{aligned} \text {either}\,\,\,\left| \frac{A_{s,\gamma _1}(v)}{A_{s,\gamma _2}(v)}\right| \le C_{18}\text { for all }x\in B_l\text { or }\left| \frac{A_{s,\gamma _2}(v)}{A_{s,\gamma _1}(v)}\right| \le C_{18}\text { for all }x\in B_l. \end{aligned}$$

(7.18)

Proof of the claim

Fix any $x_0\in \Delta _0$. Due to $|\tau _m'|_{\infty }\le C_2$ (see (4.4)), we have for any $x\in {\hat{B}}$,

$$\begin{aligned} |\tau _1(x)-\tau _2(x)|\le |\tau _1(x_0)-\tau _2(x_0)|+2C_2|x-x_0|. \end{aligned}$$

Hence there exists a constant $C({\tau _1,\tau _2})$ depending on $\tau _1,\tau _2$ such that

$$\begin{aligned}\left| \frac{A_{s,\gamma _1}(v)}{A_{s,\gamma _2}(v)}\right| \le C(\tau _1,\tau _2)\frac{f_\sigma (\gamma _1x)u(\gamma _1x)}{f_\sigma (\gamma _2x)u(\gamma _2x)}. \end{aligned}$$

For the middle term,

$$\begin{aligned}\frac{f_\sigma (\gamma _1x)}{f_\sigma (\gamma _2x)}\le 4\frac{\sup f_0}{\inf f_0}. \end{aligned}$$

Since the radius of $\gamma _2B_l$ is less than $\epsilon _2/|b|$, using Lemma 7.15, we have for every x in $B_l$

$$\begin{aligned} \frac{u(\gamma _1x)}{u(\gamma _2x)}\le \frac{\sup _{B_l}u(\gamma _1)}{\inf _{B_l}u(\gamma _2)}\le e^{2C_{17}\epsilon _2}\frac{\sup _{B_l}u(\gamma _1)}{\sup _{B_l}u(\gamma _2)}. \end{aligned}$$

Putting these together, we have

$$\begin{aligned}\left| \frac{A_{s,\gamma _1}(v)}{A_{s,\gamma _2}(v)}\right| \le C_{18}\frac{\sup _{B_l}u(\gamma _1)}{\sup _{B_l}u(\gamma _2)} \end{aligned}$$

where $C_{18}=4C(\tau _1,\tau _2)e^{2C_{17}\epsilon _2}\frac{\sup f_0}{\inf f_0}$. We have a similar inequality for $\left| \frac{A_{s,\gamma _2}(v)}{A_{s,\gamma _1}(v)}\right|$. Note that either $\frac{\sup _{B_l}u(\gamma _1)}{\sup _{B_l}u(\gamma _2)}\le 1$ or $\frac{\sup _{B_l}u(\gamma _2)}{\sup _{B_l}u(\gamma _1)}\le 1$. The proof of the claim finishes. $\square$

Now we start to compute the angle and our definitions are only for $x\in {\hat{B}}$. The function $\arg (v(\gamma _mx))$ is well defined because $|v(\gamma _mx)|\ge u(\gamma _mx)/4>0$. Let

$$\begin{aligned} \Theta (x)=b(\tau _1(x)-\tau _2(x)), \ V(x)=\arg (v(\gamma _1x))-\arg (v(\gamma _2x)), \end{aligned}$$

and let

$$\begin{aligned} \Phi (x)=\Theta (x)+V(x). \end{aligned}$$

We apply Lemma 4.5 to ${\hat{B}}$ and obtain that for $x\in {\hat{B}}$,

$$\begin{aligned}|\partial _{e_0}\Theta (x)|\ge |b|\epsilon _0, \,\,\,|\Theta '(x)|\le |b|C_2. \end{aligned}$$

For the angle function,

by (7.17) and (4.7), we have for $i\in \{1,2\}$ and $x,x'\in {\hat{B}}$

$$\begin{aligned} |\arg v(\gamma _ix)-\arg v(\gamma _ix')|&=|{\text {Im}}(\log v(\gamma _ix)-\log v(\gamma _ix'))|\\&\le \frac{|v(\gamma _ix)-v(\gamma _ix')|}{|v(\gamma _ix)|}\\&\le C_{17}|b|\frac{u(\gamma _ix)|\gamma _ix-\gamma _ix'|}{|v(\gamma _ix)|}\le |b|\epsilon _0|x-x'|/4. \end{aligned}$$

This implies that for $x,x'\in {\hat{B}}$

$$\begin{aligned} |V(x)-V(x')|\le |b|\epsilon _0|x-x'|/2. \end{aligned}$$

Combining the estimates for $\Theta$ and V, we obtain for $x,x'\in {\hat{B}}$

$$\begin{aligned} |\Phi (x)-\Phi (x')|\le b(C_2+\epsilon _0)|x-x'|, \end{aligned}$$

(7.19)

and for $x, x+te_0\in {\hat{B}}$ with $t\in {\mathbb {R}}^+$,

$$\begin{aligned}|\Phi (x)-\Phi (x+te_0)|\ge b\epsilon _0 t/2.\end{aligned}$$

Hence for $x_1=y,\ x_2=x_0$ which are the centers of $B_1$ and $B_2$ respectively, by (7.16),

$$\begin{aligned} |\Phi (x_1)-\Phi (x_2)|\in \epsilon _3[k\epsilon _0/2,(D+2)(C_2+\epsilon _0)]. \end{aligned}$$

(7.20)

Let $\epsilon _4=\epsilon _3k\epsilon _0/8.$ We claim that there exists $l\in \{1,2\}$ such that

$$\begin{aligned} d(\Phi (x_l),2\pi {\mathbb {Z}})>\epsilon _4. \end{aligned}$$

(7.21)

If not so, then both the distance from $\Phi (x_1)$ to $2\pi {\mathbb {Z}}$ and that from $\Phi (x_2)$ to $2\pi {\mathbb {Z}}$ are less than $\epsilon _4$. By (7.20) and (7.13)

$$\begin{aligned}|\Phi (x_1)-\Phi (x_2)|\le \epsilon _3(D+2)(C_2+\epsilon _0) \le 3\pi /2< 2\pi -2\epsilon _4. \end{aligned}$$

Hence $\Phi (x_1),\Phi (x_2)$ are in a ball $(2n\pi -\epsilon _4,2n\pi +\epsilon _4)$ with $n\in {\mathbb {Z}}$. This implies that

$$\begin{aligned} |\Phi (x_1)-\Phi (x_2)|\le 2\epsilon _4=\epsilon _3k\epsilon _0/4, \end{aligned}$$

contradicting with (7.20).

Without loss of generality, we may assume (7.21) holds for $x_1$. For any x in the ball $B_1$, by (7.19) and (7.10)

$$\begin{aligned}|\Phi (x)-\Phi (x_l)|\le (C_2+\epsilon _0)\epsilon _3\le k\epsilon _3\epsilon _0/16=\epsilon _4/2. \end{aligned}$$

Combined with (7.21), we have

$$\begin{aligned} d(\Phi (x),2\pi {\mathbb {Z}})\ge \epsilon _4/2. \end{aligned}$$

(7.22)

In conclusion, there exists $l\in \{1,2\}$ such that for all $x\in B_l$, $d(\Phi (x),2\pi {\mathbb {Z}})>\epsilon _4/2$ and (7.18) holds. Without loss of generality, we may assume $|A_{s,\gamma _1}(v)(x)/A_{s,\gamma _2}(v)(x)|\le C_{18}$ for all $x\in B_l$. By an elementary inequality [34, Lemma 5.12], there exists $0<\eta _0<1$ depending on $\epsilon _4$ and $C_{18}$ such that on $B_l$

$$\begin{aligned} |A_{s,\gamma _1}(v)+A_{s,\gamma _2}(v)|\le \eta _0|A_{s,\gamma _1}(v)|+|A_{s,\gamma _2}(v)|\le \eta _0A_{\sigma ,\gamma _1}(u)+A_{\sigma ,\gamma _2}(u). \end{aligned}$$

$\square$

For b with $|b|\ge 1$, let

$$\begin{aligned} \Delta _b=\left\{ x\in \Delta _0|\ d(x,\partial \Delta _0)>\frac{\epsilon _3(D+1)}{|b|}\right\} . \end{aligned}$$

(7.23)

For any $(u,v)\in {{\mathcal {C}}}_b$, we can find $\{x_i\}_{1\le i\le l_0}\subset \Lambda _0:=\Lambda _{\Gamma }\cap \Delta _0$ such that $B(x_i,\epsilon _3/|b|)$’s are disjoint balls contained in $\Delta _0$,

$$\begin{aligned}\Lambda _0\cap \Delta _b\subset \cup _{1\le i\le l_0}B(x_i,2\epsilon _3D/|b|), \end{aligned}$$

and on each $B(x_i,\epsilon _3/|b|)$ one of the $2j_0$ inequalities in Lemma 7.14 holds. In fact, suppose we have already found some points $x_i$’s but $\cup B(x_i,2\epsilon _3D/|b|)$ don’t cover the set $\Lambda _0\cap \Delta _b$. Then for a point $y\in \Lambda _0\cap \Delta _b-\cup B(x_i,2\epsilon _3D/|b|)$, we apply Lemma 7.14 to y and obtain a point $x\in B(y,\epsilon _3 D/|b|)\cap \Lambda _0$ such that Lemma 7.14 holds on $B(x,\epsilon _3/|b|)$. Moreover, the ball $B(x,\epsilon _3/|b|)$ is contained in $\Delta _0$ and it is disjoint from $\cup B(x_i,\epsilon _3/|b|)$.

Let $B_i=B(x_i,\epsilon _3/|b|)$ and $\tilde{B_i}=B(x_i,\epsilon _3/(3|b|))$ for $i=1,\ldots ,l_0$. Let $\eta \in [\eta _0,1)$ and define a $C^1$ function $\chi :\Delta _0\rightarrow [\eta ,1]$ as follows: it equals 1 outside of $\cup _{m,j,i} \gamma _{mj} B_i$; for each $B_i$, if $B_i$ is of type $\gamma _{mj}$, let $\chi (\gamma _{mj}(y))=\eta$ for $y\in \tilde{B_i}$ and $\chi \equiv 1$ on other $\gamma _{m'j'}B_i$. We can choose $\eta$ close to 1 and independent of b such that $|\chi '(x)|\le |b|$ for all $x\in \Delta _0$.

Corollary 7.24

Under the same assumptions as in Lemma 7.14, for $(u,v)\in {\mathcal {C}}_b$ and $\chi =\chi (b,u,v)$ a $C^1$ function described as above, we have

$$\begin{aligned} |{\tilde{L}}_s^{n_0} v|\le {\tilde{L}}_\sigma ^{n_0}(\chi u). \end{aligned}$$

Define $J_i=B(x_i,2\epsilon _3D/|b|)$ for $i=1,\ldots ,l_0$ and let ${\tilde{B}}=\cup \tilde{B_j}$.

Proposition 7.25

Suppose that w is a positive Lipschitz function with $|\log w(x)-\log w(y)|\le K|b||x-y|$ for some $K>0$. Then

$$\begin{aligned} \int _{{\tilde{B}}}w\mathrm {d}\nu \ge \epsilon _4\int _{\Delta _b} w\ \mathrm {d}\nu , \end{aligned}$$

(7.26)

with $\epsilon _4=\epsilon _5 e^{-4\epsilon _3DK}$, where $\epsilon _5$ comes from doubling property only depending on D and $\nu$.

Proof

Since $\cup _i J_i$ covers $\Delta _b$, it is sufficient to prove for each i we have a similar inequality. Due to hypothesis, we obtain $\inf _{\tilde{B_i}}w\ge e^{-4\epsilon _3DK}\sup _{J_i}w$. By doubling property, there exists $\epsilon _5$ depending on D such that

$$\begin{aligned}\nu (\tilde{B_i})\ge \epsilon _5\nu (J_i). \end{aligned}$$

Therefore

$$\begin{aligned}\int _{\tilde{B_i}}w\ \mathrm {d}\nu \ge \nu (\tilde{B_i})\inf _{\tilde{B_i}}w\ge \epsilon _5\nu (J_i) e^{-4\epsilon _3DK}\sup _{J_i}w\ge \epsilon _4\int _{J_i}w\ \mathrm {d}\nu . \end{aligned}$$

$\square$

7.3 Invariance of cone condition

We define the constants

$$\begin{aligned}&C_{17}'=16(\delta +\epsilon )C_2|f_0|_{\infty } |f_0^{-1}|_{\infty }+16 |f_0^{-1}|_{\infty }|f_0|_{\text {Lip}}+4C_2+2, \end{aligned}$$

(7.27)

$$\begin{aligned}&C_{17}=\max \{8|f_0^{-1}|_{\infty } |f_0|_{\text {Lip}} +(\delta +3)C_2+1+4|f_0|_{\infty }|f_0^{-1}|_{\infty }C_{17}', 6C_{16}\}. \end{aligned}$$

(7.28)

Lemma 7.29

Let $C_{17}>0$ be the constant defined in (7.28) and $n_0$ be the constant defined in (7.9). For $s=\sigma +ib$ with $|\sigma |<\epsilon$ and $|b|>1$, for $(u,v)\in {\mathcal {C}}_b$, we have

$$\begin{aligned} ({\tilde{L}}^{n_0}_{\sigma }(\chi u),\,{\tilde{L}}^{n_0}_s v)\in {\mathcal {C}}_b, \end{aligned}$$

(7.30)

where $\chi =\chi (b,u,v)$ is the same as the one in Corollary 7.24.

The proof is a verbatim of the proof of [1, Lemma 2.12].

7.4 $L^2$ contraction for bounded iterations

In this part, we will prove Proposition 7.3 for the case when m bounded by $\log |b|$. Compared with [2], where they can finish the proof of an analog of Proposition 7.3 at this stage, we have the difficulty about the boundary. More precisely, Proposition 7.25 is one of the ingredients to obtain Proposition 7.3. Now the integration region of the right hand side of (7.26) is $\Delta _b$, which is smaller than $\Delta _0$, so it just enables us to obtain $L^2$ contraction for bounded iterations. For large iteration, we will use a Lipschitz contraction lemma (Lemma 7.39) to obtain $L^2$ contraction in the next subsection.

Lemma 7.31

For $|b|>1$ and $v\in {\text {Lip}}(\Delta _0)$, if $|v|_{{\text {Lip}}}\ge C_{17} |b| |v|_{\infty }$, then

$$\begin{aligned}\Vert {\tilde{L}}_s^{n_0}v\Vert _b\le \frac{9}{10}\Vert v\Vert _b. \end{aligned}$$

Proof

We have

$$\begin{aligned}|{\tilde{L}}_s^{n_0}v|_\infty \le |v|_\infty \le \frac{1}{C_{17}|b|}|v|_{Lip}\le \frac{2}{C_{17}}\Vert v\Vert _b. \end{aligned}$$

By Lemma 7.5, we obtain

$$\begin{aligned} |{\tilde{L}}_s^{n_0}v|_{{\text {Lip}}}&\le C_{16}(1+|b|)|v|_\infty +C_{16}\lambda ^{n_0}|v|_{{\text {Lip}}}\\&\le (1+|b|)\left( \frac{C_{16}(1+|b|)}{C_{17}|b|}+C_{16}\lambda ^{n_0}\right) \Vert v\Vert _b\\&\le (1+|b|)\left( \frac{1}{3}+\frac{1}{6}\right) \Vert v\Vert _b=(1+|b|)\frac{1}{2}\Vert v\Vert _b, \end{aligned}$$

where the last inequality is due to $C_{17}\ge 6C_{16}$ and $\lambda ^{n_0}C_{17}\le 1$. $\square$

Lemma 7.32

There exist $C_{19}>0$ and $\beta <1$ such that for all $s=\sigma +ib$ with $|\sigma |<\epsilon$ and |b| large enough and $m\le [C_{19}\log |b|]$

$$\begin{aligned} \int |{\tilde{L}}^{mn_0}_{s} v|^2\mathrm {d}\nu \le \beta ^m\Vert v\Vert _b^2. \end{aligned}$$

(7.33)

Proof

If for all $0\le p\le m-1$, we have $|{\tilde{L}}_s^{pn_0}v|_{\text {Lip}}\ge C_{17} |b| |{\tilde{L}}_s^{pn_0}v|_{\infty }$, then by Lemma 7.31,

$$\begin{aligned}\int |{\tilde{L}}_s^{mn_0}v|^2\mathrm {d}\nu \le \Vert {\tilde{L}}_s^{mn_0}v\Vert _b^2\le (\frac{9}{10})^m\Vert v\Vert _b^2. \end{aligned}$$

Otherwise, suppose p is the smallest integer such that $|{\tilde{L}}_s^{pn_0}v|_{{\text {Lip}}}\le C_{17}|b||{\tilde{L}}_s^{pn_0}v|_\infty$. We consider $v'={\tilde{L}}_s^{pn_0}v$. Then Lemma 7.31 implies $\Vert v'\Vert _b\le (\frac{9}{10})^p\Vert v\Vert _b$. We only need to show that

$$\begin{aligned}\int |{\tilde{L}}_s^{(m-p)n_0}v'|^2\mathrm {d}\nu \le \beta ^{m-p}\Vert v'\Vert _b^2. \end{aligned}$$

We reduce to the case when $p=0$, that is $|v|_{{\text {Lip}}}\le C_{17}|b||v|_\infty$. Define $u_0\equiv 1,\,v_0=v/|v|_{\infty }$ and induitively,

$$\begin{aligned} u_{m+1}={\tilde{L}}^{n_0}_{\sigma }(\chi _{m}u_{m}),\,\,\, v_{m+1}={\tilde{L}}^{n_0}_{s}(v_m), \end{aligned}$$

where $\chi _m=\chi (b,u_m,v_m)$. It is immediate that $(u_0,v_0)\in {\mathcal {C}}_b$, and it follows from Lemma 7.29 that $(u_m,v_m)\in {\mathcal {C}}_b$ for all m. Hence in particular the $\chi _m$’s are well defined.

We will show that there exist $\beta _1\in (0,1)$, $\kappa >0$ and $C>0$ such that for all m

$$\begin{aligned} \int u_{m+1}^2 \mathrm {d}\nu \le \beta _1 \int u_m^2 \mathrm {d}\nu +C|b|^{-\kappa }. \end{aligned}$$

(7.34)

Then note that

$$\begin{aligned} |{\tilde{L}}^{mn_0}_s v|=|v|_{\infty } |{\tilde{L}}^{mn_0}_s v_0|=|v|_{\infty } |v_m|\le |v|_{\infty } u_m. \end{aligned}$$

As a result,

$$\begin{aligned}&\int |{\tilde{L}}^{mn_0}_s v|^2\mathrm {d}\nu \le |v|_{\infty }^2 \int u_m^2 \mathrm {d}\nu \\&\quad \le |v|^2_{\infty }(\beta _1^m \int u^2_0 \mathrm {d}\nu +C|b|^{-\kappa }\sum _{0\le l\le m-1}\beta _1^l )\\&\quad \le (\beta _1^m+C|b|^{-\kappa }/(1-\beta _1)) |v|_{\infty }^2. \end{aligned}$$

We can find $C_{19}>0$ and $\beta <1$ such that for any large enough |b|, (7.33) holds for all $m\le [C_{19}\log |b|]$.

Now we prove (7.34). By definition

$$\begin{aligned} u_{m+1}(x)&=\lambda _{\sigma }^{-n_0} f_{\sigma }^{-1}(x) \sum _{\gamma \in {{\mathcal {H}}}^{n_0}} |\gamma '(x)|^{\delta +\sigma } f_{\sigma }(\gamma x) \chi _{m}(\gamma x) u_m(\gamma x)\\&= \lambda _{\sigma }^{-n_0} f_{\sigma }^{-1}(x)\sum _{\gamma \in {{\mathcal {H}}}^{n_0}} \left( |\gamma '(x)|^{\delta /2} f_{\sigma }^{1/2}(\gamma x) u_m(\gamma x)\right) \\&\quad \left( |\gamma '(x)|^{\delta /2+\sigma } f_{\sigma }^{1/2}(\gamma x) \chi _m(\gamma x)\right) , \end{aligned}$$

so by Cauchy-Schwarz

$$\begin{aligned} u_{m+1}^2(x)&\le (\lambda _{\sigma }^{-n_0} f_{\sigma }(x))^{-2}\left( \sum _{\gamma \in {{\mathcal {H}}}^{n_0}} |\gamma '(x)|^{\delta } f_{\sigma }(\gamma x) u_{m}^2(\gamma x)\right) \\&\quad \left( \sum _{\gamma \in {{\mathcal {H}}}^{n_0}} |\gamma '(x)|^{\delta +2\sigma } f_{\sigma }(\gamma x) \chi _m^2(\gamma x)\right) \\&\le \xi (\sigma ) {\tilde{L}}^{n_0}_0 (u^2_m) {\tilde{L}}^{n_0}_{2\sigma } (\chi _m^2) \end{aligned}$$

where (noting that $\lambda _0=1$)

$$\begin{aligned} \xi (\sigma ) =(\lambda ^{-2}_{\sigma } \lambda _{2\sigma })^{n_0} \left| \frac{f_0}{f_{\sigma }}\right| _{\infty } \left| \frac{f_{2\sigma }}{f_{\sigma }}\right| _{\infty } \left| \frac{f_{\sigma }}{f_0}\right| _{\infty } \left| \frac{f_{\sigma }}{f_{2\sigma }}\right| _{\infty }. \end{aligned}$$

As in Proposition 7.25, we write $\Delta _0={\tilde{B}}\sqcup {\tilde{B}}^c$. Let ${\mathcal {H}}_c$ be the set of inverse branches given by Lemma 4.5. If $y\in {\tilde{B}}$, then there exists $\gamma _i\in {\mathcal {H}}_{c}$ such that

$$\begin{aligned} {\tilde{L}}^{n_0}_{2\sigma }(\chi ^2_{m})(y)&\le \lambda ^{-n_0}_{2\sigma } f_{2\sigma } (y)^{-1} \left( \eta ^2 |\gamma '_i(y)|^{\delta +2\sigma } f_{2\sigma }(\gamma _iy)\right. \\&\quad \left. +\sum _{\gamma \in {{\mathcal {H}}}^{n_0}\backslash \{\gamma _i\}} |\gamma '(y)|^{\delta +2\sigma } f_{2\sigma }(\gamma y) \right) \\&= {\tilde{L}}^{n_0}_{2\sigma } (1)(y)-(1-\eta ^2) \lambda ^{-n_0}_{2\sigma } f_{2\sigma }(y)^{-1} |\gamma '_i(y)|^{\delta +2\sigma }f_{2\sigma }(\gamma _iy)\\&\le 1-(1-\eta ^2) 2^{-(n_0+2)} \inf f_0 \cdot |f_0|_{\infty }^{-1} \cdot \inf _{\{\gamma _i\in {\mathcal {H}}_c\}} |\gamma '_i|^{\delta +2\sigma }=\eta _1<1, \end{aligned}$$

In this way we obtain that there exists $\eta _1<1$ such that

$$\begin{aligned} u^2_{m+1}(y)\le {\left\{ \begin{array}{ll} \eta _1 \xi (\sigma ) {\tilde{L}}^{n_0}_0 (u_m^2)(y), &{} y\in {\tilde{B}},\\ \xi (\sigma ) {\tilde{L}}^{n_0}_0 (u_m^2)(y), &{} y\in {\tilde{B}}^c. \end{array}\right. } \end{aligned}$$

Since $(u_m,v_m)\in {\mathcal {C}}_b$, it follows in particular that $|\log u_m|_{\text {Lip}}\le C_{17} |b|$. Hence by (7.9),

$$\begin{aligned} u_m^2(\gamma x)/u_m^2(\gamma y)\le \exp (2C_{17}\lambda ^{n_0} |b|d(x,y))\le \exp (2|b|d(x,y)). \end{aligned}$$

Let $w={\tilde{L}}^{n_0}_0(u_m^2)$. Then

$$\begin{aligned} \frac{w(x)}{w(y)}&=\frac{f_0(y)\sum _{\gamma \in {{\mathcal {H}}}^{n_0}}|\gamma '(x)|^{\delta } f_0(\gamma x)u_m^2(\gamma x) }{f_0(x) \sum _{\gamma \in {{\mathcal {H}}}^{n_0}} |\gamma '(y)|^{\delta } f_0(\gamma y) u_m^2(\gamma y)}\\&\le \exp \left( \left( 2|f_0^{-1}|_{\infty } |f_0|_{\text {Lip}} +\delta C_2 +2|b|\right) d(x,y)\right) . \end{aligned}$$

Hence $|\log w|_{\text {Lip}}\le K|b|$ with $K=2|f_0^{-1}|_{\infty } |f_0|_{\text {Lip}} +\delta C_2+2$. Using Proposition 7.25, we have

$$\begin{aligned} (1-\eta _1) \int _{{\tilde{B}}} w\mathrm {d}\nu \ge \epsilon _4 (1-\eta _1)\int _{\Delta _b} w \mathrm {d}\nu . \end{aligned}$$

Setting $\beta '=1-\epsilon _4(1-\eta _1)$, we can further write

$$\begin{aligned} \eta _1\int _{{\tilde{B}}} w\mathrm {d}\nu +\int _{\Delta _b-{\tilde{B}}} w\mathrm {d}\nu \le \beta ' \int _{\Delta _b} w\mathrm {d}\nu \le \beta '\int _{\Delta _0} w \mathrm {d}\nu . \end{aligned}$$

Hence

$$\begin{aligned} \int _{\Delta _b} u^2_{m+1}\mathrm {d}\nu&\le \xi (\sigma ) \left( \eta _1\int _{{\tilde{B}}} {\tilde{L}}^{n_0}_0(u^2_m)\mathrm {d}\nu +\int _{\Delta _b-{\tilde{B}}} {\tilde{L}}^{n_0}_0 (u^2_m)\mathrm {d}\nu \right) \nonumber \\&\le \xi (\sigma ) \beta ' \int _{\Delta _0} {\tilde{L}}^{n_0}_0(u^2_m)\mathrm {d}\nu =\xi (\sigma ) \beta ' \int _{\Delta _0} u^2_m\mathrm {d}\nu . \end{aligned}$$

(7.35)

By (5.30), (7.23) and $|u_{m+1}|\le 1$,

$$\begin{aligned} \int _{\Delta _0-\Delta _b}u_{m+1}^2 \mathrm {d}\nu \le \nu (\Delta _0-\Delta _b)\le C|b|^{-\kappa }. \end{aligned}$$

(7.36)

Finally we can shrink $\epsilon$ if necessary so that $\xi (\sigma )\beta '<1$ for $|\sigma |<\epsilon$ and then (7.35) and (7.36) imply (7.34). $\square$

7.5 Proof of Proposition 7.3

Lemma 7.37

There exist $\epsilon \in (0,1),\, \tau \in (0,1)$ and $C_{20}>0$ such that for all $s=\sigma +ib$ with $|\sigma |<\epsilon$, $n\ge 1$ and $v\in {\text {Lip}}(\Delta _0)$, we have

$$\begin{aligned} |{\tilde{L}}^n_s v|^2_{\infty } \le C_{20} (1+|b|)\tau ^n |v|_{\infty } \Vert v\Vert _b+C_{20}B^n|v|_{\infty } \int |v| \mathrm {d}\nu \end{aligned}$$

where $B>1$ is a constant depending on $\epsilon$ and it tends to 1 as $\epsilon \rightarrow 0$.

Proof

We have

$$\begin{aligned} |{\tilde{L}}^n_s v(x)|&\le \lambda _{\sigma }^{-n} f_{\sigma }^{-1}(x) \sum _{\gamma \in {{\mathcal {H}}}^n} |\gamma '(x)|^{\delta +\sigma } f_{\sigma }(\gamma x) |v|(\gamma x)\\&= \lambda ^{-n}_{\sigma } f_{\sigma }^{-1}(x) \sum _{\gamma \in {{\mathcal {H}}}^n} \left( |\gamma '(x)|^{\delta /2+\sigma } f_{\sigma }^{1/2}(\gamma x) |v|^{1/2}(\gamma x)\right) \\&\quad \left( |\gamma '(x)|^{\delta /2} f^{1/2}_{\sigma }(\gamma x) |v|^{1/2}(\gamma x)\right) . \end{aligned}$$

Using Cauchy-Schwarz, we obtain

$$\begin{aligned} |{\tilde{L}}^n_s v(x)|^2 \le (\lambda _{\sigma }^{-2}\lambda _{2\sigma })^n \xi (\sigma ) {\tilde{L}}^n_{2\sigma }(|v|) (x) \cdot {\tilde{L}}^n_0(|v|)(x), \end{aligned}$$

where $\xi (\sigma )=|f_0/f_{\sigma }|_{\infty } |f_{2\sigma }/f_{\sigma }|_{\infty } |f_{\sigma }/f_0|_{\infty } |f_{\sigma }/f_{2\sigma }|_{\infty }\le 64$. Hence

$$\begin{aligned} |{\tilde{L}}^n_s v|^2_{\infty } \le 64 B^n |v|_{\infty } |{\tilde{L}}^n_0(|v|)|_{\infty }, \end{aligned}$$

(7.38)

where $B>1$ is a constant depending on $\epsilon$ with $B\rightarrow 1$ as $\epsilon \rightarrow 0$.

Since ${\tilde{L}}_0$ is a normalized transfer operator for the uniformly expanding map T, there exists $\tau _1\in (0,1)$ such that $|{\tilde{L}}^n_0 v|_{\infty }\le C\tau _1^n\Vert v\Vert _{\text {Lip}}$ for all $v\in \text {Lip}(\Delta _0)$ with $\int v \mathrm {d}\nu =0$. (This is a consequence of spectral gap of quasi-compact operator ${\tilde{L}}_0$.) Hence by decomposing |v| into $(|v|-\int |v|\mathrm {d}\nu )+\int |v|\mathrm {d}\nu$, we obtain

$$\begin{aligned} | {\tilde{L}}^n_0(|v|)|_{\infty } \le 2C\tau _1^n \Vert v\Vert _{\text {Lip}}+\int |v|\mathrm {d}\nu . \end{aligned}$$

Substituting into (7.38), we have

$$\begin{aligned} |{\tilde{L}}^n_s v|_{\infty }^2\le 128 C(B\tau _1)^n (1+|b|)|v|_{\infty } \Vert v\Vert _b +64 B^n |v|_{\infty } \int |v| \mathrm {d}\nu . \end{aligned}$$

Finally, shrink $\epsilon$ if necessary so that $\tau =B\tau _1<1$. $\square$

Lemma 7.39

There exist $C>0,\ \epsilon \in (0,1),\, A>0$ and $\beta \in (0,1)$ such that

$$\begin{aligned} \Vert {\tilde{L}}^{mn_0}_s v\Vert _b\le C\beta ^m \Vert v\Vert _b \end{aligned}$$

for all $m\ge A\log |b|,\, s=\sigma +ib$ with $|\sigma |<\epsilon$ and |b| large enough, and all $v\in {\text {Lip}}(\Delta _0)$.

Proof

Let $N=[C_{19}\log |b|]n_0$. Using Lemma 7.37 for ${\tilde{L}}_s^N v$ and $n=lN$, Lemma 7.7 and (7.33), we obtain

$$\begin{aligned} |{\tilde{L}}_s^{(l+1)N}v |_\infty ^2&\le C_{20}(1+|b|)\tau ^{lN}|{\tilde{L}}_s^N v|_\infty \Vert {\tilde{L}}_s^Nv\Vert _b\\&\quad +C_{20}B^{lN}|{\tilde{L}}_s^Nv|_\infty (\int |{\tilde{L}}_s^Nv|^2\mathrm {d}\nu )^{1/2}.\\&\le 2C_{16}C_{20}(1+|b|)\tau ^{lN}|v|_\infty \Vert v\Vert _b+2C_{16}C_{20}B^{lN}|v|_\infty \beta ^{N/2}\Vert v\Vert _b. \end{aligned}$$

We fix l depending on $\tau , C_{19}$ and $n_0$ such that $(1+|b|)\tau ^{lN/2}\le 1$. Then by shrinking B if necessary, there exists $\beta _1<1$, such that

$$\begin{aligned} |{\tilde{L}}_s^{(l+1)N}v|_\infty \le \beta _1^{(l+1)N}\Vert v\Vert _b. \end{aligned}$$

(7.40)

For Lipschitz norm, we have

$$\begin{aligned}&|{\tilde{L}}_s^{(l+2)N}v|_{{\text {Lip}}}\\&\quad \le C_{16}(1+|b|)|{\tilde{L}}_s^{(l+1)N}v|_\infty +C_{16}\lambda ^N|{\tilde{L}}_s^{(l+1)N}v|_{{\text {Lip}}}\\&\quad \le C_{16}(1+|b|)\beta _1^{(l+1)N}\Vert v\Vert _b+C_{16}^2\lambda ^N((1+|b|)|v|_\infty +\lambda ^{(l+1)N}|v|_{{\text {Lip}}})\\&\quad \le C_{16}^2(1+|b|)\Vert v\Vert _b(\beta _1^{(l+1)N}+\lambda ^N+\lambda ^{(l+2)N})\le 3C_{16}^2(1+|b|)\beta _2^N\Vert v\Vert _b, \end{aligned}$$

for some $\beta _2<1$, where we use Lemma 7.5 to get the first inequality and (7.40) to get the second one. For the infinity norm, by (7.40) and Lemma 7.7, we obtain

$$\begin{aligned} |{\tilde{L}}_s^{(l+2)N}v|_\infty \le 2C_{16}\beta _1^{(l+1)N}\Vert v\Vert _b. \end{aligned}$$

Combining these two norm estimates, we obtain

$$\begin{aligned} \Vert {\tilde{L}}_s^{(l+2)N}v\Vert _b\le C_{16}^2(2\beta _1^{(l+1)N}+3\beta _2^{N})\Vert v\Vert _b\le \beta _3^{(l+2)N/n_0}\Vert v\Vert _b, \end{aligned}$$

(7.41)

for some $\beta _3<1$ if |b| is large enough to absorb the constant $6C_{16}^2$.

Let $A=2(l+2)C_{19}$ and $N_1=(l+2)N/n_0=(l+2)[C_{19}\log |b|]\le A\log |b|$. For $m\ge A\log |b|$, we can write $m=dN_1+r$ with $r\in {\mathbb {N}}$ and $r< N_1$. Therefore by (7.41) and Lemma 7.7,

$$\begin{aligned} \Vert {\tilde{L}}_s^{mn_0}v\Vert _b= & {} \Vert {\tilde{L}}_s^{dN_1n_0}({\tilde{L}}_s^{rn_0}v)\Vert _b\le \beta _3^{dN_1}\Vert {\tilde{L}}_s^{rn_0}v\Vert _b\\&\quad \le 2C_{16}\beta _3^{dN_1}\Vert v\Vert _b\le 2C_{16}(\sqrt{\beta _3})^m\Vert v\Vert _b. \end{aligned}$$

$\square$

Proof of Proposition 7.3

It is sufficient to prove that for all $m\in {\mathbb {N}}$,

$$\begin{aligned} \int |{\tilde{L}}_s^{mn_0}v|^2\mathrm {d}\nu \le C\beta ^m\Vert v\Vert ^2_b. \end{aligned}$$

(7.42)

Then for any $k\in {\mathbb {N}}$, suppose $k=mn_0+r$ with $0\le r<n_0$. We have

$$\begin{aligned} \int |L_s^kv|^2\mathrm {d}\mu _E\le & {} C\lambda _\sigma ^{2k}\int |{\tilde{L}}_s^k(f_{\sigma }^{-1}v)|^2\mathrm {d}\nu \le C\lambda _\sigma ^{2k} \beta ^m\Vert {\tilde{L}}_s^r(f_{\sigma }^{-1}v)\Vert _b^2\\\le & {} C\lambda _\sigma ^{2k}\beta ^m\Vert f_{\sigma }^{-1}v\Vert _b^2\le C\lambda _{\sigma }^{2k}\beta ^m \Vert v\Vert _b^2. \end{aligned}$$

By choosing $\epsilon$ small such that $\lambda _\sigma ^{2n_0}\beta <1$ for any $|\sigma |<\epsilon$, we obtain Proposition 7.3.

It remains to prove (7.42). For $m> A\log |b|$, by Lemma 7.39, we obtain

$$\begin{aligned} \int |{\tilde{L}}_s^{mn_0}v|^2\mathrm {d}\nu \le \Vert {\tilde{L}}_s^{mn_0}v \Vert _b^2\le C\beta ^m\Vert v\Vert _b^2. \end{aligned}$$

For $A\log |b|\ge m\ge C_{19}\log |b|$, by (7.33) and Lemma 7.7, we know

$$\begin{aligned}&\int |{\tilde{L}}_s^{mn_0}v|^2\mathrm {d}\nu \le \beta ^{[C_{19}\log |b|]}\Vert {\tilde{L}}_s^{(m-[C_{19}\log |b|])n_0}v \Vert _b^2\\&\quad \le 2C_{16}\beta ^{[C_{19}\log |b|]}\Vert v\Vert ^2_b\le 2C_{16}\beta _1^m\Vert v\Vert ^2_b \end{aligned}$$

for some $\beta _1=\beta ^{C_{19}/A}<1$.

The case when $m\le C_{19}\log |b|$ has been verified in Lemma 7.32. $\square$

8 Exponential mixing

In this section, we prove Theorem 4.13. As a first step, an analogous result concerning expanding semiflow will be proved. Let $T:\Lambda _+\rightarrow \Lambda _+$ be the uniformly expanding map and $R:\Lambda _+\rightarrow {\mathbb {R}}_+$ be the roof function as defined in Proposition 4.1. Set $\Lambda _+^{R}=\{(x,t)\in \Lambda _+\times {\mathbb {R}}: 0\le t<R(x)\}$. We define a semi-flow $T_t:\Lambda _+^R\rightarrow \Lambda _+^R$ by $T_s(x,t)=(T^nx, t+s-R_n(x))$ where n is the unique integer satisfying $R_n(x)\le t+s<R_{n+1}(x)$. Recall that $\nu$ is the unique T-invariant ergodic probability measure on $\Lambda _+$. Then the flow $T_t$ preserves the probability measure $\nu ^{R}=\nu \times {\text {Leb}}/(\nu \times {\text {Leb}})(\Lambda _+^R)$. We will also use the probability measure $\mu _E^{R}=\mu _E\times {\text {Leb}}/(\mu _E\times {\text {Leb}})(\Lambda _+^R)$ on $\Lambda _+^R$. We show that $T_t$ is exponentially mixing.

For a bounded function on $\Lambda _+^R$, we define two norms. Set

$$\begin{aligned}&\Vert U\Vert _{{{\mathcal {B}}}_0}=|U|_\infty +\sup _{ (x,a)\ne (x',a')\in \Lambda _+^R}\frac{|U(x,a)-U(x',a')|}{d(x,x')+|a-a'|},\\&\Vert V\Vert _{{{\mathcal {B}}}_1}=|V|_\infty +\sup _{x\in \Lambda _+} \frac{{\text {Var}}_{(0,R(x))}\{t\mapsto V(x,t) \}}{R(x)}, \end{aligned}$$

where ${\text {Var}}_{(0,R(x))}\{t\mapsto V(x,t) \}$ is the total variation of the function $t\mapsto V(x,t)$ on the interval (0, R(x)).

Theorem 8.1

There exist $C>0,\ \epsilon >0$ such that for all $t>0$ and for any two functions $U,\ V$ on $\Lambda _{+}^R$ with $\Vert U\Vert _{{{\mathcal {B}}}_0},\ \Vert V\Vert _{{{\mathcal {B}}}_1}$ finite, we have

$$\begin{aligned} \left| \int U\cdot V\circ T_t\mathrm {d}\mu _E^R-\left( \int U\mathrm {d}\mu _E^R\right) \left( \int V\mathrm {d}\nu ^R\right) \right| \le Ce^{-\epsilon t}\Vert U\Vert _{{{\mathcal {B}}}_0}\Vert V\Vert _{{{\mathcal {B}}}_1}. \end{aligned}$$

Remark 8.2

Applying this theorem to the function $U(x,t)\frac{\mathrm {d}\nu ^R}{\mathrm {d}\mu _E^R}(x)$, we obtain

$$\begin{aligned} \left| \int U\cdot V\circ T_t\mathrm {d}\nu ^R-\left( \int U\mathrm {d}\nu ^R\right) \left( \int V\mathrm {d}\nu ^R\right) \right| \le Ce^{-\epsilon t}\Vert U\Vert _{{{\mathcal {B}}}_0}\Vert V\Vert _{{{\mathcal {B}}}_1}.\nonumber \\ \end{aligned}$$

(8.3)

With Proposition 7.3 available, Theorem 8.1 can be proved essentially along the same lines as the proof of [2, Theorem 7.3] (see also [2, Section 7.5]). We provide a sketch of the proof here. For a pair of functions U, V, let $\rho (t)=\int U\cdot V\circ T_t\mathrm {d}\mu _E^R$ be the correlation function and the observation is that the Laplace transform of $\rho$, denoted by $\hat{\rho }$, can be expressed as a sum of twisted transfer operators $L_s$ [2, Lemma 7.17]. One shows that $\hat{\rho }$ admits an analytic continuation to a neighborhood of each point $s=ib$ and this part of the argument uses the quasi-compactness of the twisted transfer operators [2, Lemma 7.21, 7.22]. When |b| is large, the Dolgopyat-type estimate (Proposition 7.3), which is a replacement of [2, Proposition 7.7] in the current setting, is used to imply that $\hat{\rho }$ admits an analytic extension to a strip $\{s=\sigma +ib\in {\mathbb {C}}: |\sigma |<\sigma _0\}$ for all sufficiently small $\sigma _0$ [2, Corollary 7.20]. The result of exponential mixing then follows from the classical Paley-Wiener theorem [2, Theorem 7.23].

The difference between our result and that in [2] is the classes of functions in concern. The only adjustment we need to make is [2, Lemma 7.18], which is a norm estimate for $C^1$ functions in their paper, but for functions with finite ${{\mathcal {B}}}_0$ norm in the current setting. The precise statement is as follows. For a function $U:\Lambda _+\rightarrow {\mathbb {R}}$ with $\Vert U\Vert _{{{\mathcal {B}}}_0}<\infty$ and $s\in {\mathbb {C}}$, set ${\hat{U}}_{s}(x)=\int _0^{R(x)} e^{-ts}U(x,t)\mathrm {d}t$.

Lemma 8.4

There exists $C>0$ such that for $s=\sigma +ib$ with $|\sigma |\le \epsilon _o/4$ ($\epsilon _o$ is given as in Proposition 4.1 (5)), the function $L_s{\hat{U}}_{-s}$ is Lipschitz on $\Delta _0$ and

$$\begin{aligned} \Vert L_s{\hat{U}}_{-s}\Vert _b\le \frac{C\Vert U\Vert _{{{\mathcal {B}}}_0}}{\max \{ 1,|b|\}}. \end{aligned}$$

Proof

We first prove for $x\in \Lambda _{+}$ we have

$$\begin{aligned} |{\hat{U}}_{-s}(x)|\le \frac{Ce^{\epsilon _oR(x)/2}}{\max \{1,|b|\}}\Vert U\Vert _{{{\mathcal {B}}}_0}. \end{aligned}$$

By definition, we have

$$\begin{aligned} {\hat{U}}_{-s}(x)=\int _0^{R(x)}U(x,t)e^{ts}\mathrm {d}t. \end{aligned}$$

The case when $|b|\le 1$ is easy. When $|b|>1$, one uses integration by parts and the fact that U is Lipschitz with respect to t to obtain

$$\begin{aligned} |{\hat{U}}_{-s}(x)|\le (2|U|_\infty e^{\epsilon _oR(x)/4}+|U|_{{\text {Lip}}}R(x)e^{\epsilon _o R(x)/4})/\max \{1,|b|\}. \end{aligned}$$

Then

$$\begin{aligned} |L_s{\hat{U}}_{-s}|\le \frac{C\Vert U\Vert _{{{\mathcal {B}}}_0}}{\max \{1,|b|\}}L_\sigma (e^{\epsilon _oR/2}). \end{aligned}$$

Observe that by (4.2)

$$\begin{aligned} L_\sigma (e^{\epsilon _oR/2})=\sum _{\gamma \in {{\mathcal {H}}}}|\gamma '(x)|^{\delta +\sigma }e^{\epsilon _o R(\gamma x)/2}\le \sum _{\gamma \in {{\mathcal {H}}}}|\gamma '(x)|^{\delta -3\epsilon _o/4}<\infty .\nonumber \\ \end{aligned}$$

(8.5)

So $|L_s{\hat{U}}_{-s}|\le \frac{C\Vert U\Vert _{{{\mathcal {B}}}_0}}{\max \{1,|b|\}}$.

We estimate the Lipschitz norm of $L_s{\hat{U}}_{-s}$. We have

$$\begin{aligned}&L_s{\hat{U}}_{-s}(x)-L_s{\hat{U}}_{-s}(y)=\sum _{\gamma \in {{\mathcal {H}}}}|\gamma '(x)|^{\delta +s}({\hat{U}}_{-s}(\gamma x)-{\hat{U}}_{-s}(\gamma y))\\&\quad +(|\gamma '(x)|^{\delta +s}-|\gamma '(y)|^{\delta +s}){\hat{U}}_{-s}(\gamma y). \end{aligned}$$

The term $|\gamma '(x)|^{\delta +s}-|\gamma '(y)|^{\delta +s}$ can be estimated using Proposition 4.1 (4). For the term ${\hat{U}}_{-s}(\gamma x)-{\hat{U}}_{-s}(\gamma y)$, suppose that $R(\gamma x)\ge R(\gamma y)$, we use Proposition 4.1 (4) again and get

$$\begin{aligned} |{\hat{U}}_{-s}(\gamma x)-{\hat{U}}_{-s}(\gamma y)|&\le |R(\gamma x)-R(\gamma y)||U|_\infty e^{\sigma R(\gamma x)}\\&\quad +\int _0^{R(\gamma y)}|U(\gamma x,t)-U(\gamma y,t)|e^{t\sigma }\mathrm {d}t\\&\le (C_1 e^{\sigma R(\gamma x)}+R(\gamma y)e^{\sigma R(\gamma y)})\Vert U\Vert _{{{\mathcal {B}}}_0}d(x,y). \end{aligned}$$

Then we use (8.5) to conclude that there exists some C (independent of U) such that

$$\begin{aligned} |L_s{\hat{U}}_{-s}|_{{\text {Lip}}}\le C \Vert U\Vert _{{{\mathcal {B}}}_0}. \end{aligned}$$

$\square$

Proof of Theorem 4.13

Now Theorem 4.13 can be proved using the same lines as the proof of [2, Theorem 2.7] (see also [2, Section 8.2]). In particular, in the proof of [2, Lemma 8.3], we use (8.3) to replace [2, Theorem 7.3] and Proposition 4.11 (2) to relate the measures $\hat{\nu }^{R}$ and $\nu ^R$. $\square$

9 Resonance-free region

Recall that $\Gamma$ is a geometrically finite discrete subgroup in $G={\text {SO}}(d+1,1)^{\circ }$. We begin by defining the measures $m^{{\text {BR}}}$, $m^{{\text {BR}}_{*}}$ and $m^{{\text {Haar}}}$. Recall the definition of the BMS measure on ${\text {T}}^1({\mathbb {H}}^{d+1})\cong \partial ^2({\mathbb {H}}^{d+1})\times {\mathbb {R}}$:

$$\begin{aligned} \mathrm {d}{\tilde{m}}^{{\text {BMS}}}(x,x_-,s)=e^{\delta \beta _x(o,x_*)} e^{\delta \beta _{x_-}(o,x_*)} \mathrm {d}\mu (x) \mathrm {d}\mu (x_-)\mathrm {d}s, \end{aligned}$$

where $x_*$ is the based point of the unit tangent vector given by $(x,x_-,s)$. We define the measures ${\tilde{m}}^{{\text {BR}}}$, ${\tilde{m}}^{{\text {BR}}_{*}}$ and ${\tilde{m}}^{{\text {Haar}}}$ on ${\text {T}}^1({\mathbb {H}}^{d+1})\cong \partial ^2({\mathbb {H}}^{d+1})\times {\mathbb {R}}$ similarly as follows:

$$\begin{aligned} \mathrm {d}{\tilde{m}}^{{\text {BR}}}(x,x_-,s)&=e^{d\beta _x(o,x_*)} e^{\delta \beta _{x_-}(o,x_*)} \mathrm {d}m_o(x) \mathrm {d}\mu (x_-)\mathrm {d}s;\\ \mathrm {d}{\tilde{m}}^{{\text {BR}}_{*}}(x,x_-,s)&=e^{\delta \beta _x(o,x_*)} e^{\delta \beta _{x_-}(o,x_*)} \mathrm {d}\mu (x) \mathrm {d}m_o(x_-)\mathrm {d}s;\\ \mathrm {d}{\tilde{m}}^{{\text {Haar}}}(x,x_-,s)&=e^{d \beta _x(o,x_*)} e^{d\beta _{x_-}(o,x_*)} \mathrm {d}m_o(x) \mathrm {d}m_o(x_-)\mathrm {d}s, \end{aligned}$$

where $m_o$ is the unique probability measure on $\partial ({\mathbb {H}}^{d+1})$ which is invariant under the stabilizer of o in G.

These measures are all left $\Gamma$-invariant and induce measures on ${\text {T}}^1(\Gamma \backslash {\mathbb {H}}^{d+1})$, which we will denote by $m^{{\text {BR}}}$, $m^{{\text {BR}}_{*}}$ and $m^{{\text {Haar}}}$ respectively. Here we do not normalize the BMS measure to a probability measure, which is different from the previous part.

By [36, Theorem 5.8], Theorem 1.1 implies exponential decay of matrix coefficients.

Theorem 9.1

There exists $\eta >0$ such that for any compactly supported functions $\phi , \psi \in C^1({\text {T}}^1(M))$, we have

$$\begin{aligned} e^{(d-\delta )t}\int _{{\text {T}}^1(M)} \phi \cdot \psi \circ {{\mathcal {G}}}_t\ \mathrm {d}m^{{\text {Haar}}}=\frac{m^{{\text {BR}}_{*}} (\phi ) m^{{\text {BR}}} (\psi )}{m^{{\text {BMS}}}({\text {T}}^1(M))}+O(\Vert \phi \Vert _{C^1} \Vert \psi \Vert _{C^1}e^{-\eta t}) \end{aligned}$$

for all $t>0$, where O depends on the supports of $\phi ,\psi$.

For $x,y\in {\mathbb {H}}^{d+1}$ and $T>0$, let

$$\begin{aligned} N(T,x,y)=\#\{\gamma \in \Gamma \,|\,d(x,\gamma y)\le T \}, \end{aligned}$$

where d is the hyperbolic distance on ${\mathbb {H}}^{d+1}$. In [33], it was shown that Theorem 9.1 implies the following:

Corollary 9.2

There exists $\eta >0$ such that for any $x,y\in {\mathbb {H}}^{d+1}$ and $T>0$, we have

$$\begin{aligned} N(T,x,y)=c_{x,y}e^{\delta T}+O(e^{(\delta -\eta )T}), \end{aligned}$$

where $c_{x,y}>0$ is a constant depending on x, y.

Proof of Corollary 1.2

For $x,y\in {\mathbb {H}}^{d+1}$ and $s\in {\mathbb {C}}$ with $\Re s>\delta$, let $P_s(x,y)$ be the Poincaré series defined by

$$\begin{aligned} P_s(x,y)=\sum _{\gamma \in \Gamma } e^{-sd(x,\gamma y)}. \end{aligned}$$

We first prove that $P_s(x,y)$ is meromorphic on $\Re s>\delta -\eta$ with a unique pole $s=\delta$. By Fubini’s theorem

$$\begin{aligned} P_s(x,y)= & {} \int _{0}^\infty \frac{1}{s}e^{-sT}N(T,x,y)\mathrm {d}T=\int _{0}^\infty \frac{1}{s}e^{-(s-\delta )T}c_{x,y}\mathrm {d}T\\&\quad +\int _{0}^\infty \frac{1}{s}e^{-sT}(N(T,x,y)-c_{x,y}e^{\delta T})\mathrm {d}T. \end{aligned}$$

The first part is a meromorphic function of s having a unique pole at $s=\delta$. The second part, it follows from Corollary 9.2 that it is absolutely convergence if $\Re s>\delta -\eta$, hence it is analytic on $\Re s>\delta -\eta$. Then we use [21, Theorem 7.3] to deduce that the resolvent $R_M(s)$ is also analytic on $\{s\in {\mathbb {C}}:\, \delta -\eta<\Re s<\delta \}$. $\square$

Appendix: Proof of Lemma 6.9

Proof of Lemma 6.9

We divide into the cases when r lies in different intervals. Let $\beta =C\sqrt{\eta }$.

• Case A: $r\le \eta h(p)$ By Lemma 5.7, we have $|(\gamma ^{-1})'x|=h(p)/d(x,p)^2$. Using Lemma 6.7, we have

  $$\begin{aligned} N_r(\partial J_p)\subset B(p, 2\eta h(p))-B(p,c_{4}\eta h(p)). \end{aligned}$$

  (10.1)

  Hence for $x\in N_r(\partial J_p)$

  $$\begin{aligned} |(\gamma ^{-1})'x|\in [1/(4\eta ^2h(p)),1/(c_{4}^2\eta ^2 h(p))]. \end{aligned}$$

  Then

  $$\begin{aligned} N_{r/(4\eta ^2h(p))} (\partial \gamma ^{-1}J_p)\subset \gamma ^{-1}N_r(\partial J_p) \subset N_{r/(c^2_4\eta ^2h(p))}(\partial \gamma ^{-1}J_p).\qquad \end{aligned}$$

  (10.2)

  Notice that $\partial \gamma ^{-1} J_p=\partial (B_Y(2/\eta )\times R_{p,\eta })$ and $R_{p,\eta }$ is a parallelotope tiled by the translations of ${{\overline{\Delta }}}_0'$.

  • Case $A_1$: $r\le \eta ^2h(p)$

    Recall that Lemma 5.29 is proved using Lemma 5.31. Using the same argument, we obtain an analog of Lemma 5.29 for $N_r(\partial \gamma ^{-1} J_p)$. Using this version of Lemma 5.29 with $\epsilon =4\beta /c_{4}^2$, the inequality $r/(4\eta ^2 h(p))< 1$ and (10.2), we have

    $$\begin{aligned}&\mu ( \gamma ^{-1}N_{\beta r}(\partial J_p))\le \mu (N_{\beta r/(c_{4}^2 \eta ^2 h(p))}(\partial \gamma ^{-1}J_p))\\&\quad \le \lambda \mu (N_{r/(4\eta ^2 h(p))}(\partial \gamma ^{-1}J_p))\le \lambda \mu ( \gamma ^{-1}N_r(\partial J_p)). \end{aligned}$$

    Using Lemma 5.17, we obtain

    $$\begin{aligned} \frac{\mu (N_{\beta r}(\partial J_p))}{\mu (N_r(\partial J_p))}\le C\frac{\mu (\gamma ^{-1}N_{\beta r}(\partial J_p))}{\mu (\gamma ^{-1} N_{r}(\partial J_p))} \le C\lambda , \end{aligned}$$

    where $\lambda$ tends to zero as $\eta$ tends to zero.

  • Case $A_2$: $\eta ^{3/2} h(p)<r\le \eta h(p)$

    We compute the measure by counting the number of translations of $\Delta _0$. By (10.1) and Lemma 5.7, we obtain $\gamma ^{-1}N_r(\partial J_p)\subset B(p',1/(c_{4}\eta ))-B(p',1/(2\eta ))$. Let $\gamma ' \Delta _0$ be any fundamental domain contained in $\gamma ^{-1}N_r(\partial J_p)$ with $\gamma '\in \Gamma _{\infty }$. Using Lemma 5.18, we obtain that $\mu (\gamma '\Delta _0)\approx \eta ^{2\delta }\mu (\Delta _0)$. By Lemma 5.17, (10.2)

    $$\begin{aligned} \mu (N_{\beta r}(\partial J_p))\ll h(p)^{\delta }\mu (N_{\beta r/(c_{4}^2 \eta ^2 h(p))}(\partial \gamma ^{-1}J_p)). \end{aligned}$$

    By counting the number of fundamental domains, we obtain that the region $N_{\beta r/(c_{4}^2 \eta ^2 h(p))}(\partial \gamma ^{-1}J_p)$ can be covered by $(1/\eta )^{k-1} \cdot (\beta r/(c_{4}^2 \eta ^2 h(p)))$ disjoint rectangles $\gamma '\Delta _{0}$ with $\gamma '\in \Gamma _{\infty }$. So we have

    $$\begin{aligned} \mu (N_{\beta r}(\partial J_p))\ll h(p)^{\delta }\cdot (1/\eta )^{k-1} \cdot (\beta r/(c_{4}^2 \eta ^2 h(p)))\cdot \eta ^{2\delta }\mu (\Delta _0). \end{aligned}$$

    By Lemma 5.17, (10.2)

    $$\begin{aligned} \mu (N_r(\partial J_p))\gg h(p)^{\delta }\mu (N_{r/(4\eta ^2 h(p))}(\partial \gamma ^{-1}J_p)). \end{aligned}$$

    Meanwhile, as $r/(4\eta ^2h(p))\ge 1/4\eta ^{1/2}$, the number of rectangles $\gamma '\Delta _{0}$ inside $N_{r/(4\eta ^2 h(p))}(\partial \gamma ^{-1}J_p)$ is greater than $(1/\eta )^{k-1}\cdot (r/(4\eta ^2 h(p)))$. Hence we have

    $$\begin{aligned} \mu (N_r(\partial J_p)) \gg h(p)^{\delta }\cdot (1/\eta )^{k-1}\cdot (r/(4\eta ^2 h(p)))\cdot \eta ^{2\delta }\mu (\Delta _0). \end{aligned}$$

    Therefore,

    $$\begin{aligned} \mu (N_{\beta r}(\partial J_p)\ll \beta \mu (N_r(\partial J_p)). \end{aligned}$$

• Case B: $\eta ^{1/2} h(p)\le r\le h(p)$

  We handle this case using (5.16). By Lemma 6.7 and the inequality $h(p)/2>\beta r\ge \eta h(p)$, we have

  $$\begin{aligned}&\mu (N_{\beta r}(\partial J_p))\le \mu (J_p\cup N_{\beta r}(\partial J_p))\le \mu (B(p,\eta h(p)+\beta r))\\&\quad \le \mu (B(p,2\beta r))\ll (2\beta r)^{2\delta -k}h(p)^{k-\delta }. \end{aligned}$$

  Meanwhile, we have

  $$\begin{aligned} \mu (J_p\cup N_r (\partial J_p))\ge \mu (B(p,r))\gg r ^{2\delta -k}h(p)^{k-\delta }. \end{aligned}$$

  Hence

  $$\begin{aligned}&\frac{\mu (N_r(\partial J_p)-N_{\beta r}(\partial J_p))}{\mu (N_{\beta r}(\partial J_p))}=\frac{\mu (J_p\cup N_r(\partial J_p))-\mu (J_p\cup N_{\beta r}(\partial J_p))}{\mu (N_{\beta r}(\partial J_p))} \\&\quad \gg \frac{r ^{2\delta -k}h(p)^{k-\delta }-(2\beta r)^{2\delta -k}h(p)^{k-\delta } }{(2\beta r)^{2\delta -k}h(p)^{k-\delta }}. \end{aligned}$$

  Therefore

  $$\begin{aligned} \mu (N_{\beta r}(\partial J_p))\ll \beta ^{2\delta -k} \mu (N_r(\partial J_p)). \end{aligned}$$

Now to prove (6.10) we consider $\eta ^{1/2} r$ and r. Then one of them belongs to $(0,\eta ^2 h(p))\cup [ \eta ^{3/2} h(p),\eta h(p)]\cup [\eta ^{1/2} h(p),h(p)]$. Inequality (6.10) follows from the observation that

$$\begin{aligned} \frac{\mu (N_{C\eta r}(\partial J_p))}{\mu (N_r(\partial J_p))} \le \min \left\{ \frac{\mu (N_{C\eta r}(\partial J_p))}{\mu (N_{\beta r}(\partial J_p))}, \frac{\mu (N_{\beta r}(\partial J_p))}{\mu (N_r(\partial J_p))}\right\} . \end{aligned}$$

$\square$