Good Inducing Schemes for Uniformly Hyperbolic Flows, and Applications to Exponential Decay of Correlations

Abstract

Given an Axiom A attractor for a $C^{1+\alpha }$ flow ($\alpha >0$), we construct a countable Markov extension with exponential return times in such a way that the inducing set is a smoothly embedded unstable disk. This avoids technical issues concerning irregularity of boundaries of Markov partition elements and enables an elementary approach to certain questions involving exponential decay of correlations for SRB measures.

1 Introduction

Statistical properties of Anosov and Axiom A diffeomorphisms [3, 33] were developed extensively in the 1970 s. Key tools were the construction of finite Markov partitions [10, 32] and the spectral properties of transfer operators [28]. In particular, ergodic invariant probability measures were constructed corresponding to any Hölder potential [12, 29, 31]; moreover, it was shown that hyperbolic basic sets for Axiom A diffeomorphisms are always exponentially mixing up to a finite cycle for such measures, see for example [12, 22, 29].

Still in the 1970 s, finite Markov partitions were constructed [11, 26] for Anosov and Axiom A flows. This allows us to model each hyperbolic basic set as a suspension flow over a subshift of finite type, enabling the study of thermodynamic formalism (see e.g. [14]) and statistical properties (see e.g. [17, 27]). By the Anosov alternative [3, 23], a transitive Anosov flow is mixing if and only if it is a constant suspension over an Axiom A diffeomorphism.

However, rates of mixing for Axiom A flows are still poorly understood. By [24, 30], mixing Axiom A flows can mix arbitrarily slowly. Although there has been important progress starting with [16, 18, 20], it remains an open question whether mixing Anosov flows have exponential decay of correlations. Very recently, this question was answered positively [35] in the case of $C^\infty$ three-dimensional flows.

It turns out that using finite Markov partitions for flows raises technical issues due to the irregularity of their boundaries [5, 15, 34]. Even in the discrete-time setting, it is known that the boundaries of elements of a finite Markov partition need not be smooth [13]. In this paper, we propose using the approach of [36] to circumvent such issues at least in the case of SRB measures. In particular, we show that

Any attractor¹for an Axiom A flow can be modeled by a suspension flow over a full branch countable Markov extension where the inducing set is a smoothly embedded unstable disk. The roof function, though unbounded, has exponential tails.

A precise statement is given in Theorem 2.1 below.

Remark 1.1

The approach of Young towers [36] has proved to be highly effective for studying discrete-time examples like planar dispersing billiards and Hénon-like attractors where suitable Markov partitions are not available. However, as shown in the current paper, there can be advantages (at least in continuous time) to working with countable Markov extensions even when there is a well-developed theory of finite Markov partitions. The extra flexibility of Markov extensions can be used not only to construct the extension but to ensure good regularity properties of the partition elements.

As a consequence of Theorem 2.1, we obtain an elementary proof of the following result:

Theorem 1.2

Suppose that $\Lambda$ is an Axiom A attractor with SRB measure $\mu$ for a $C^{1+}$ flow $\phi _t$ with $C^{1+}$ stable holonomies² and such that the stable and unstable bundles are not jointly integrable. Then for all Hölder observables $v,\,w:\Lambda \rightarrow {{\mathbb {R}}}$, there exist constants $c,C>0$ such that

$$\begin{aligned} \Big |\int _\Lambda v\,w\circ \phi _t\,d\mu -\int _\Lambda v\,d\mu \int _\Lambda w\,d\mu \Big |\le C e^{-c t} \quad \text {for all }t>0. \end{aligned}$$

Remark 1.3

Joint nonintegrability holds for an open and dense set of Axiom A flows and their attractors, see [19] and references therein. It implies mixing and is equivalent to mixing for codimension one Anosov flows. It is conjectured to be equivalent to mixing for Anosov flows [23].

Remark 1.4

• (a) In the case when the unstable direction is one-dimensional and the stable holonomies are $C^2$, this result is due to [4, 5, 8, 9]. In particular, using the fact that stable bunching is a robust sufficient condition for smoothness of stable holonomies together with the robustness of joint nonintegrability, [4] constructed the first robust examples of Axiom A flows with exponential decay of correlations. The smoothness condition on stable holonomies was relaxed from $C^2$ to $C^{1+}$ in [6] extending the class of examples in [4]. This class of examples is extended further by Theorem 1.2 with the removal of the one-dimensionality restriction on unstable manifolds.

• (b) There is no restriction on the dimension of unstable manifolds in [8], and it is not surprising that the smoothness assumption on stable holonomies can also be relaxed as in [6]. However, there is a crucial hypothesis in [8] on the regularity of the inducing set in the unstable direction which is nontrivial in higher dimensions.

Theorem 1.2 is stated in the special case of Anosov flows in [15]. In [15, Appendix] it is argued that at least in the Anosov case the Markov partitions of [26] are sufficiently regular that the methods in [8] can be pushed through. In [5], a sketch is given of how to prove Theorem 1.2 also in the Axiom A case, but the details are not fully worked out.

As mentioned, our approach in this paper completely bypasses such issues since our inducing set is a smoothly embedded unstable disk. Moreover, our method works equally well for Anosov flows and Axiom A attractors. As a consequence, we recover the examples in [15], in particular that codimension one volume-preserving mixing $C^{1+}$ Anosov flows are exponentially mixing in dimension four and higher.

The remainder of the paper is organized as follows. In Sect. 2, we state precisely and prove our result on good inducing for attractors of Axiom A flows. In Sect. 3, we prove a result on exponential mixing for a class of skew product Axiom A flows, extending/combining the results in [6, 8]. In Sect. 4, we complete the proof of Theorem 1.2.

2 Good Inducing for Attractors of Axiom A Flows

Let $\phi _t: M\rightarrow M$ be a $C^{1+}$ flow defined on a compact Riemannian manifold $(M,d_M)$, and let $\Lambda \subset M$ be a closed $\phi _t$-invariant subset. We assume that $\Lambda$ is an attracting transitive uniformly hyperbolic set with adapted norm and that $\Lambda$ is not a single trajectory. In particular, there is a continuous $D\phi _t$-invariant splitting $T_\Lambda M=E^s\oplus E^c\oplus E^u$ where $E^c$ is the one-dimensional central direction tangent to the flow, and there exists $\lambda \in (0,1)$ such that $|D\phi _t v|\le \lambda ^t|v|$ for all $v\in E^s$, $t\ge 1$; $|D\phi _{-t} v|\le \lambda ^t|v|$ for all $v\in E^u$, $t\ge 1$. Since the time-s map $\phi _s:\Lambda \rightarrow \Lambda$ is ergodic for all but countably many choices of $s\in {{\mathbb {R}}}$ [25], we can scale time by a constant close to one if necessary so that $\phi _{-1}:\Lambda \rightarrow \Lambda$ is transitive. Then there exists $p\in \Lambda$ such that $\bigcup _{i\ge 1}\phi _{-i}p$ is dense in $\Lambda$.

We can define (local) stable disks $W^s_\delta (y)=\{z\in W^s(y):d_M(y,z)<\delta \}$ for $\delta >0$ sufficiently small for all $y\in \Lambda$. Define local center-stable disks $W^{cs}_\delta (y)=\bigcup _{|t|<\delta }\phi _t W^s_\delta (y)$. Let ${\text {Leb}}$ and d denote induced Lebesgue measure and induced distance on local unstable manifolds. It is convenient to define local unstable disks $W^u_\delta (y)=\{z\in W^u(y):d(y,z)<\delta \}$ using the induced distance.

For $\delta _0$ small, define ${{\mathcal {D}}}=W^u_{\delta _0}(p)$ and ${\widehat{{\mathcal {D}}}}=\bigcup _{x\in {{\mathcal {D}}}}W^{cs}_{\delta _0}(x)$. Define $\pi :{\widehat{{\mathcal {D}}}}\rightarrow {{\mathcal {D}}}$ such that $\pi | W^{cs}_{\delta _0}(x)\equiv x$. Whenever $\phi _ny\in {\widehat{{\mathcal {D}}}}$, we set $g_ny=\pi (\phi _ny)$.

We are now in a position to give a precise description of our inducing scheme.

Theorem 2.1

There exists an open unstable disk $Y=W^u_\delta (p)\subset {{\mathcal {D}}}$ (for some $\delta \in (0,\delta _0)$) and a discrete return time function $R:Y\rightarrow {{\mathbb {Z}}}^+\cup \{\infty \}$ such that

• (i) ${\text {Leb}}(R>n)=O(\gamma ^n)$ for some $\gamma \in (0,1)$;

• (ii) Each connected component of $\{R=n\}$ is mapped by $\phi _n$ into ${\widehat{{\mathcal {D}}}}$ and mapped homeomorphically by $g_n$ onto Y.

Remark 2.2

Let ${{\mathcal {P}}}$ be the partition of Y consisting of connected components of $\{R=n\}$ for $n\ge 1$. (It follows from Theorem 2.1(i) that ${{\mathcal {P}}}$ is a partition of $Y\bmod 0$.) Define $F:Y\rightarrow Y$, $F=g_R=\pi \circ \phi _R$. Note that F is locally the composition of a time-R map $\phi _R$ (where R is constant on each partition element) with a center-stable holonomy. Since center-stable holonomies are Hölder continuous, it follows that F maps partition elements $U\in {{\mathcal {P}}}$ homeomorphically onto Y and that $F|_U:U\rightarrow Y$ is a bi-Hölder bijection. If moreover, the center-stable holonomies are $C^1$, then the partition elements are diffeomorphic to disks (in contrast to the situation for finite Markov partitions of $\Lambda$ [13]).

In the remainder of this section, we prove Theorem 2.1. Our proof is essentially the same as in [36, Section 6] for Axiom A diffeomorphisms, but we closely follow the treatment in [2] which provides many of the details of arguments sketched in [36].

Choice of constants We can choose $\delta _0>0$ such that the following bounded distortion property holds³: there exists $C_1\ge 1$ so that

$$\begin{aligned} \frac{|\det D\phi _n(x)|E^u|}{|\det D\phi _n(y)|E^u|} \le C_1 \end{aligned}$$

(2.1)

for every $n\ge 1$ and all $x,y\in \Lambda$ with $\phi _nx,\phi _ny$ in the same unstable disk such that $d(\phi _jx,\phi _jy)<4\delta _0$ for all $0\le j \le n$.

By standard results about stable holonomies, $\pi$ is absolutely continuous and $C^\alpha$ for some $\alpha \in (0,1)$ when restricted to unstable disks in ${\widehat{{\mathcal {D}}}}$. For $\delta _0$ sufficiently small, there exists $C_2$, $C_3\ge 1$ such that

$$\begin{aligned} C_2^{-1}\le \frac{{\text {Leb}}(\pi (E))}{{\text {Leb}}(E)} \le C_2 \end{aligned}$$

(2.2)

for all Lebesgue-measurable subset $E\subset W^u_{\delta _0}(y)\cap {\widehat{{\mathcal {D}}}}$ and all $y\in \Lambda$, and

$$\begin{aligned} d(\pi x,\pi y)\le C_3 d(x,y)^\alpha \end{aligned}$$

(2.3)

for all $x,y\in {\widehat{{\mathcal {D}}}}$ with x, y in the same unstable disk such that $d(x,y)<4\delta _0$.

Let $d_u=\dim E^u$ and fix $L>3$ so that

$$\begin{aligned} C_1C_2^2\frac{2^{d_u}-1}{(L-1)^{d_u}}<\frac{1}{4}. \end{aligned}$$

(2.4)

By the local product structure, there exists $\delta _1\in (0,\delta _0)$ such that $W^{cs}_{\delta _0}(x)\cap W^u_{\delta _0}(y)$ consists of precisely one point for all $x,y\in \Lambda$ with $d_M(x,y)<4\delta _1$. Similarly, there exists $\delta \in (0,\delta _1)$ such that $W^{cs}_{\delta _1}(x)\cap W^u_{\delta _1}(y)$ consists of precisely one point for all $x,y\in \Lambda$ with $d_M(x,y)<(L+1)\delta$. Since local center-stable/unstable manifolds lie in the corresponding cones, and the center-stable/unstable cones are uniformly transverse, the intersection point $z\in W^{cs}_{\delta _1}(x)\cap W^u_{\delta _1}(y)$ satisfies $d(z,y)\le C_4d_M(x,y)$ where $C_4\ge 1$ is a constant. Shrink $\delta >0$ if necessary so that $C_3(3\delta )^\alpha <\frac{1}{2}\delta _0$ and $C_4(L+1)\delta <\delta _0$. Choose $N_1\ge 1$ such that $\bigcup _{i=1}^{N_1} \phi _{-i}p$ is $\delta$-dense in $\Lambda$.

Construction of the partition We consider various small neighborhoods ${{\mathcal {D}}}_c=W^u_{c\delta }(p)$ with $c\in \{1,2,L-1,L\}$. Define ${\widetilde{{\mathcal {D}}}}_c=\bigcup _{x\in {{\mathcal {D}}}_c}W^{cs}_{\delta _1}(x)$.

Take $Y={{\mathcal {D}}}_1$. Define a partition $\{I_k:k\ge 1\}$ of ${{\mathcal {D}}}_2\setminus {{\mathcal {D}}}_1$,

$$\begin{aligned} I_k=\big \{y\in {{\mathcal {D}}}_2: \delta (1+\lambda ^{\alpha k}) \le d(y,p) < \delta (1+\lambda ^{\alpha (k-1)})\big \}. \end{aligned}$$

Fix $\varepsilon >0$ small (as stipulated in Propositions 2.4 and 2.5 and Lemma 2.9 below). We define sets $Y_n$ and functions $t_n:Y_n\rightarrow {{\mathbb {N}}}$, and $R:Y\rightarrow {{\mathbb {Z}}}^+$ inductively, with $Y_n=\{R> n\}$. Define $Y_0=Y$ and $t_0\equiv 0$. Inductively, suppose that $Y_{n-1}=Y\setminus \{R<n\}$ and that $t_{n-1}:Y_{n-1}\rightarrow {{\mathbb {N}}}$ is given. Write $Y_{n-1}=A_{n-1}\,{{\dot{\cup }}}\,B_{n-1}$ where

$$\begin{aligned} A_{n-1}=\{t_{n-1}=0\}, \qquad B_{n-1}=\{t_{n-1}\ge 1\}. \end{aligned}$$

Consider the neighborhood

$$\begin{aligned} A^{(\varepsilon )}_{n-1}=\big \{y\in Y_{n-1}: d(\phi _ny,\phi _nA_{n-1})<\varepsilon \big \} \end{aligned}$$

of the set $A_{n-1}$. Define $U_{nj}^L$, $j\ge 1$, to be the connected components of $A^{(\varepsilon )}_{n-1}\cap \phi _{-n}{\widetilde{{\mathcal {D}}}}_L$ that are mapped inside ${\widetilde{{\mathcal {D}}}}_L$ by $\phi _n$ and mapped homeomorphically onto ${{\mathcal {D}}}_L$ by $g_n$. Let

$$\begin{aligned} U_{nj}^c=U_{nj}^L\cap g_n^{-1}{{\mathcal {D}}}_c \quad \text {for c=1,2,L-1.} \end{aligned}$$

Define $R|U_{nj}^1=n$ for each $U_{nj}^1$ and take $Y_n=Y_{n-1}\setminus \bigcup _j U_{nj}^1$. Finally, define $t_n:Y_n\rightarrow {{\mathbb {N}}}$ as

$$\begin{aligned} t_n(y) = {\left\{ \begin{array}{ll} k, &{} \quad y\in \bigcup _j U_{nj}^2 \text { and } g_ny\in I_k \, \text {for some }k\ge 1\\ 0, &{} \quad y\quad \in A_{n-1} \setminus \bigcup _j U_{nj}^2 \\ t_{n-1}(y)-1, &{} \quad y\in B_{n-1} \setminus \bigcup _j U_{nj}^2 \end{array}\right. } \end{aligned}$$

and take $A_n=\{t_n=0\}$, $B_n=\{t_n\ge 1\}$ and $Y_n = A_n\,{{\dot{\cup }}}\,B_n$.

Remark 2.3

By construction, property (ii) of Theorem 2.1 is satisfied. It remains to verify that ${\text {Leb}}(R>n)$ decays exponentially.

Visualisation of $B_n$. The set $B_n$ is a disjoint union $B_n=\bigcup _{m=1}^n C_n(m)$ where $C_n(m)$ is a disjoint union of collars around each component of $\{R= m\}$. Each collar in $C_n(m)$ is homeomorphic under $g_m$ to $\bigcup _{k\ge n-m+1}I_k$ with outer ring homeomorphic under $g_m$ to $I_{n-m+1}$, and the union of outer rings is the set $\{t_n=1\}$. This picture presupposes Proposition 2.4 below which guarantees that each new generation of collars $C_n(n)$ does not intersect the set $\bigcup _{1\le m\le n-1}C_{n-1}(m)$ of collars in the previous generation. A sample visualization after 7 generations is shown in Fig. 1.

Fig. 1

Visualisation of Y after 7 generations where there is one return at time 2 and one return at time 7. The pink region $B_7$ consists of collars around the sets $\{R=2\}$ and $\{R=7\}$ that have made a return. The two outermost shells $\{t_7=1\}$ and $\{t_7=2\}$ of each collar are shown. The collars in $B_7$ are diffeomorphic by $g_2$ and $g_7$, respectively, to an annulus; in reality the collar around $\{R=2\}$ should be slightly distorted and the collar around $\{R=7\}$ more so (and smaller)

We now turn to the proof of Proposition 2.4. A cautionary diagram is shown in Fig. 2.

Fig. 2

This diagram indicates some subtleties in the proof of Proposition 2.4. The solid black curve denotes part of the boundary of $A_{n-1}$ formed by a much earlier return. The dashed black curve denotes the corresponding part of the boundary of $A_{n-1}^{(\varepsilon )}$. The red curve denotes the boundary of a $U_{nj}^L$. Even though $U_{nj}^L$ is contained in $A_{n-1}^{(\varepsilon )}$ it is not clear that $U_{nj}^{L-1}$ is contained in $A_{n-1}$

Proposition 2.4

Choose $\varepsilon <(C_3^{-1}\delta )^{1/\alpha }$ sufficiently small that $W^u_\varepsilon (x)\subset {\widehat{{\mathcal {D}}}}$ for all $x\in {\widetilde{{\mathcal {D}}}}_L$. Then $\bigcup _jU_{nj}^{L-1}\subset A_{n-1}$ for all $n\ge 1$.

Proof

We argue by contradiction. There is nothing to prove for $n=1$. Let $n\ge 2$ be least such that the result fails and choose j such that $U_{nj}^{L-1}$ intersects $B_{n-1}$. Then either (i) $U_{nj}^{L-1}\subset B_{n-1}$, or (ii) $U_{nj}^{L-1}$ intersects $\partial A_{n-1}$.

In case (i), choose $x\in U_{nj}^{L-1}$ (so in particular $\phi _nx\in {\widehat{{\mathcal {D}}}}$) with $g_nx=p$. Since $U_{nj}^{L-1}\subset U_{nj}^L\subset A_{n-1}^{(\varepsilon )}$, there exists $y\in A_{n-1}$ with $d(\phi _nx,\phi _ny)<\varepsilon$. In particular, $\phi _ny\in {\widehat{{\mathcal {D}}}}$ so $g_ny$ is well-defined. Note that $x\in U_{nj}^{L-1}$ and $y\not \in U_{nj}^{L-1}$ since $U_{nj}^{L-1}\subset B_{n-1}$. Hence the geodesic $\ell$ in ${{\mathcal {D}}}$ joining $g_nx$ and $g_ny$ intersects $g_n\partial U_{nj}^{L-1}$. Choose $z\in \partial U_{nj}^{L-1}\cap g_n^{-1}\ell$. Since $g_n=\pi \circ \phi _n$, it follows from (2.3) that

$$\begin{aligned} \delta<(L-1)\delta =d(g_nx,g_nz)< d(g_nx,g_ny)\le C_3 d(\phi _nx,\phi _ny)^\alpha<C_3\varepsilon ^\alpha <\delta \end{aligned}$$

which is a contradiction. This rules out case (i).

In case (ii), choose $x\in U_{nj}^{L-1}\cap \partial A_{n-1}$. We show below that there exists $y\in \partial A_{n-1}^{(\varepsilon )}$ such that $d(\phi _nx,\phi _ny)\le \varepsilon$. In particular, $g_nx$ and $g_ny$ are well-defined and $d(g_nx,g_ny)\le C_3\varepsilon ^\alpha <\delta$. Since $U_{nj}^L\subset A_{n-1}^{(\varepsilon )}$, we have that $y\not \in U_{nj}^L$. It follows that $g_nx\in {{\mathcal {D}}}_{L-1}$ while $g_ny\not \in {{\mathcal {D}}}_L$. Hence $d(g_nx,g_ny)\ge \delta$ which is the desired contradiction.

It remains to verify that there exists $y\in \partial A_{n-1}^{(\varepsilon )}$ such that $d(\phi _nx,\phi _ny)\le \varepsilon$. Since n is least, $B_{n-1}$ is a disjoint union of collars as described in the visualization above. Hence there exists a collar $Q\subset C_{n-1}(n-k)$ intersected by $U_{nj}^{L-1}$ for some $1\le k<n$ such that x lies in the outer boundary $\partial _oQ$ of Q. Note that $\partial _o Q= \partial A_{n-1}\cap Q$. Let D denote the disk enclosed by $\partial _oQ$ and let

$$\begin{aligned} S=D\cap \partial (\phi _{-n}B_\varepsilon (\phi _n\partial D)). \end{aligned}$$

We claim that $S\ne \emptyset$ and $S\subset Q$. Then S is a $(\dim Y-1)$-dimensional sphere contained in $\partial A_{n-1}^{(\varepsilon )}$ and there exists $y\in S$ with the desired properties, see Fig. 3. (The point of the claim is that S lies entirely in $Y_{n-1}$.)

Fig. 3

Schematic of the claim in the proof of Proposition 2.4. The pink region is a collar Q. The blue curve denotes the boundary of a $U_{nj}^{L-1}$. The claim is that the relevant part S of $\partial A_{n-1}^{(\varepsilon )}$ lies inside Q enabling the choice of a point y close to, but not too close to, x

Note that $g_{n-k}$ maps Q homeomorphically onto the set $J=\bigcup _{i\ge k}I_i$ which is an annulus of radial thickness $\delta \lambda ^{\alpha k}$. By (2.3), $\phi _{n-k}$ maps Q homeomorphically onto a set ${{\tilde{J}}}=\pi ^{-1}J$ of radial thickness at least $(C_3^{-1}\delta \lambda ^{\alpha k})^{1/\alpha } = (C_3^{-1}\delta )^{1/\alpha }\lambda ^k$.

Moreover, $\phi _k({{\tilde{J}}}\cap \phi _{n-k}A_{n-1}^{(\varepsilon )})\subset \phi _nA_{n-1}^{(\varepsilon )}$ is contained in the set of points within d-distance $\varepsilon$ of $\phi _n\partial A_{n-1}^{(\varepsilon )}$, so by definition of $\lambda$ we have that ${{\tilde{J}}}\cap \phi _{n-k}A_{n-1}^{(\varepsilon )}$ is contained in the set of points within d-distance $\varepsilon \lambda ^k$ of the outer boundary of ${{\tilde{J}}}$. Since $\varepsilon <(C_3^{-1}\delta )^{1/\alpha }$, we obtain that ${{\tilde{J}}}\cap \phi _{n-k}\partial A_{n-1}^{(\varepsilon )}$ is homeomorphic to a $(\dim Y-1)$-dimensional sphere contained entirely inside ${{\tilde{J}}}$. Hence $S=Q\cap \partial A_{n-1}^{(\varepsilon )}$ is homeomorphic to a $(\dim Y-1)$-dimensional sphere contained entirely inside Q, as required.  $\square$

Proposition 2.5

Choose $\varepsilon <\big \{C_3^{-1}\delta (\lambda ^{-\alpha }-1)\big \}^{1/\alpha }$. Then for all $n\ge 1$,

• (a) $A_{n-1}^{(\varepsilon )}\subset \{y\in Y_{n-1}:t_{n-1}(y)\le 1\}$ for all $n\ge 1$.

• (b) $\phi _{-n}W_\varepsilon ^u(\phi _nx)\subset A_{n-1}^{(\varepsilon )}$ for all $x\in A_{n-1}$.

Proof

(a) Suppose that $t_{n-1}(y)>1$. Then there exists a collar in $C_{n-1}(n-k)$ containing y. Let Q denote the outer ring of the collar with outer boundary $Q_1$ and inner boundary $Q_2$. Then $t_{n-1}|Q\equiv 1$ and $t_{n-1}(y)>1$, so y lies inside the region bounded by $Q_2$.

Suppose for contradiction that $y\in A_{n-1}^{(\varepsilon )}$. Then we can choose $x\in A_{n-1}$ with $d(\phi _nx,\phi _ny)<\varepsilon$. Let $\ell$ be the geodesic in $W^u_\varepsilon (\phi _nx)$ connecting $\phi _nx$ to $\phi _ny$ and define $q_j\in Q_j\cap \phi _{-n}\ell$ for $j=1,2$.

Recall that Q is homeomorphic under $g_{n-k}$ to $I_k$. Moreover, $g_{n-k}q_j$ lie in distinct components of the boundary of $I_k$, so

$$\begin{aligned} d(g_{n-k}q_1,g_{n-k}q_2)\ge \delta (\lambda ^{\alpha (k-1)}-\lambda ^{\alpha k})=\delta (\lambda ^{-\alpha }-1)\lambda ^{\alpha k}. \end{aligned}$$

Hence

$$\begin{aligned} \begin{aligned} d(\phi _nq_1,\phi _nq_2)&\ge \lambda ^{-k} d(\phi _{n-k}q_1,\phi _{n-k}q_2)\\&\ge \lambda ^{-k}\big \{C_3^{-1} d(g_{n-k}q_1,g_{n-k}q_2)\big \}^{1/\alpha }\ge \big \{C_3^{-1}\delta (\lambda ^{-\alpha }-1)\big \}^{1/\alpha }>\varepsilon . \end{aligned} \end{aligned}$$

But $d(\phi _nq_1,\phi _nq_2)\le d(\phi _ny,\phi _nx)<\varepsilon$ so we obtain the desired contradiction.

(b) Let $x\in A_{n-1}$ and $y\in \phi _{-n}W^u_\varepsilon (\phi _nx)$. Note that $y\in A_{n-1}^{(\varepsilon )}$ if and only if $y\in Y_{n-1}$. Hence we must show that $y\in Y_{n-1}$. If not, then there exists $k\ge 1$ such that $y\in \{R=n-k\}$. Define $Q\subset C_{n-1}(n-k)$ to be the outer ring of the corresponding collar. Choosing $q_1$ and $q_2$ as in part (a) we again obtain a contradiction. $\square$

Lemma 2.6

There exists $a_1>0$ such that for all $n\ge 1$,

• (a) ${\text {Leb}}(B_{n-1}\cap A_n)\ge a_1{\text {Leb}}(B_{n-1})$.

• (b) ${\text {Leb}}(A_{n-1}\cap B_n)\le \tfrac{1}{4}{\text {Leb}}(A_{n-1})$.

• (c) ${\text {Leb}}(A_{n-1}\cap \{R=n\})\le \tfrac{1}{4}{\text {Leb}}(A_{n-1})$.

Proof

(a) Let $y\in B_{n-1}$. By Proposition 2.4, $y\not \in \bigcup _j U_{nj}^{L-1}$ so in particular $y\in Y_n$. Note that $t_n(y)=0$ if and only if $t_{n-1}(y)=1$. Hence $B_{n-1}\cap A_n=\{t_{n-1}=1\}$.

Now let $Q\subset C_{n-1}(n-k)\subset B_{n-1}$ be a collar ($1\le k\le n$) with outer ring $Q\cap A_n=Q\cap \{t_{n-1}=1\}$. Then $g_{n-k}=\pi \circ \phi _{n-k}$ maps Q homeomorphically onto $\bigcup _{i\ge k}I_i$ and $Q\cap \{t_{n-1}=1\}$ homeomorphically onto $I_k$. Let $d_u=\dim E^u$. By (2.1) and (2.2),

$$\begin{aligned} \begin{aligned} \frac{{\text {Leb}}(Q)}{{\text {Leb}}(Q\cap A_n)}&= \frac{{\text {Leb}}(Q)}{{\text {Leb}}(Q\cap \{t_{n-1}=1\})} \le C_1\frac{{\text {Leb}}(\phi _{n-k}Q)}{{\text {Leb}}(\phi _{n-k}(Q\cap \{t_{n-1}=1\}))}\\&\le C_1C_2^2\frac{{\text {Leb}}(\bigcup _{i\ge k}I_i)}{{\text {Leb}}(I_k)} = C_1C_2^2D(d_u,\lambda ^\alpha ,k) \end{aligned} \end{aligned}$$

where $D(d_u,\lambda ,k)=\dfrac{(1+\lambda ^{k-1})^{d_u}-1}{(1+\lambda ^{k-1})^{d_u}-(1+\lambda ^k)^{d_u}}$. Since $\lim _{k\rightarrow \infty }D(d_u,\lambda ,k)=(1-\lambda )^{-1}$, we obtain that ${\text {Leb}}(Q)\le C_1C_2^2D{\text {Leb}}(Q\cap A_n)$ where $D=\sup _{k\ge 1}$$D(d_u,\lambda ^\alpha ,k)$ is a constant depending only on $d_u$ and $\lambda ^\alpha$. Summing over collars Q, it follows that ${\text {Leb}}(B_{n-1})\le C_1C_2^2D {\text {Leb}}(B_{n-1}\cap A_n)$.

(b) By Proposition 2.4, $U_{nj}^2\subset U_{nj}^{L-1}\subset A_{n-1}$ for each j. It follows that $A_{n-1}\cap B_n=\bigcup _j U_{nj}^2\setminus U_{nj}^1$. By (2.1), (2.2) and (2.4),

$$\begin{aligned} \frac{{\text {Leb}}(U_{nj}^2\setminus U_{nj}^1)}{{\text {Leb}}(U_{nj}^{L-1})}\le C_1 C_2^2 \frac{{\text {Leb}}({{\mathcal {D}}}_2\setminus {{\mathcal {D}}}_1)}{{\text {Leb}}({{\mathcal {D}}}_{L-1})}=C_1C_2^2\frac{2^{d_u}-1}{(L-1)^{d_u}}<\frac{1}{4}. \end{aligned}$$

Hence

$$\begin{aligned} \frac{{\text {Leb}}(A_{n-1}\cap B_n)}{{\text {Leb}}(A_{n-1})}\le \frac{\sum _j{\text {Leb}}(U_{nj}^2\setminus U_{nj}^1)}{\sum _j{\text {Leb}}(U_{nj}^{L-1})} <\frac{1}{4}. \end{aligned}$$

(c) Proceeding as in part (b) with $U_{nj}^2\setminus U_{nj}^1$ replaced by $U_{nj}^1$, leads to the estimate

$$\begin{aligned} \frac{{\text {Leb}}(A_{n-1}\cap \{R=n\})}{{\text {Leb}}(A_{n-1})}\le \frac{\sum _j{\text {Leb}}(U_{nj}^1)}{\sum _j{\text {Leb}}(U_{nj}^{L-1})}\le \frac{C_1C_2^2}{(L-1)^{d_u}}<\frac{1}{4}. \end{aligned}$$

$\square$

Corollary 2.7

For all $n\ge 1$,

• (a) ${\text {Leb}}(A_{n-1}\cap A_n)\ge \tfrac{1}{2}{\text {Leb}}(A_{n-1})$.

• (b) ${\text {Leb}}(B_{n-1}\cap B_n)\le (1-a_1){\text {Leb}}(B_{n-1})$.

• (c) ${\text {Leb}}(B_n)\le \frac{1}{4}{\text {Leb}}(A_{n-1})+(1-a_1){\text {Leb}}(B_{n-1})$.

• (d) ${\text {Leb}}(A_n)\ge \frac{1}{2}{\text {Leb}}(A_{n-1})+a_1{\text {Leb}}(B_{n-1})$.

Proof

Recall that $A_{n-1}\subset Y_{n-1}= Y_n\,{{\dot{\cup }}}\,\{R=n\} =A_n\,{{\dot{\cup }}}\, B_n\,{{\dot{\cup }}}\,\{R=n\}$. Hence by Lemma 2.6(b,c),

$$\begin{aligned} \begin{aligned} {\text {Leb}}(A_{n-1})&={\text {Leb}}(A_{n-1}\cap A_n) +{\text {Leb}}(A_{n-1}\cap B_n) +{\text {Leb}}(A_{n-1}\cap \{R=n\})\\&\le {\text {Leb}}(A_{n-1}\cap A_n)+\tfrac{1}{2}{\text {Leb}}(A_{n-1}), \end{aligned} \end{aligned}$$

proving (a). Similarly, by Lemma 2.6(a),

$$\begin{aligned} \begin{aligned} {\text {Leb}}(B_{n-1})&={\text {Leb}}(B_{n-1}\cap A_n) +{\text {Leb}}(B_{n-1}\cap B_n) +{\text {Leb}}(B_{n-1}\cap \{R=n\})\\&\ge a_1{\text {Leb}}(B_{n-1})+{\text {Leb}}(B_{n-1}\cap B_n), \end{aligned} \end{aligned}$$

proving (b).

Next, recall that $B_n=B_n\cap Y_{n-1}= B_n\cap \big (A_{n-1}\,{{\dot{\cup }}}\, B_{n-1}\big )$. Hence part (c) follows from Lemma 2.6(b) and part (b). Similarly, $A_n=A_n\cap \big (A_{n-1}\,{{\dot{\cup }}}\, B_{n-1}\big )$ and part (d) follows from Lemma 2.6(a) and part (a). $\square$

Corollary 2.8

There exists $a_0>0$ such that ${\text {Leb}}(B_n)\le a_0{\text {Leb}}(A_n)$ for all $n\ge 0$.

Proof

Let $a_0=\dfrac{2+a_1}{2a_1}.$ We prove the result by induction. The case $n=0$ is trivial since $B_0=\emptyset$. For the induction step from $n-1$ to n, we consider separately the cases ${\text {Leb}}(B_{n-1})>\frac{1}{2a_1}{\text {Leb}}(A_{n-1})$ and ${\text {Leb}}(B_{n-1})\le \frac{1}{2a_1}{\text {Leb}}(A_{n-1})$.

Suppose first that ${\text {Leb}}(B_{n-1})>\frac{1}{2a_1}{\text {Leb}}(A_{n-1})$. By Corollary 2.7(c),

$$\begin{aligned} {\text {Leb}}(B_n)< \big \{\tfrac{1}{2} a_1+(1-a_1)\big \}{\text {Leb}}(B_{n-1}) = (1-\tfrac{1}{2} a_1){\text {Leb}}(B_{n-1}) < {\text {Leb}}(B_{n-1}). \end{aligned}$$

By Corollary 2.7(d),

$$\begin{aligned} {\text {Leb}}(A_n)> (\tfrac{1}{2}+a_1\tfrac{1}{2a_1}){\text {Leb}}(A_{n-1})={\text {Leb}}(A_{n-1}). \end{aligned}$$

Hence by the induction hypothesis,

$$\begin{aligned} {\text {Leb}}(B_n)<{\text {Leb}}(B_{n-1})\le a_0{\text {Leb}}(A_{n-1})<a_0{\text {Leb}}(A_n), \end{aligned}$$

establishing the result at time n.

Finally, suppose that ${\text {Leb}}(B_{n-1})\le \frac{1}{2a_1}{\text {Leb}}(A_{n-1})$. By Corollary 2.7(a,c),

$$\begin{aligned} \begin{aligned} {\text {Leb}}(B_n)&\le \tfrac{1}{4}{\text {Leb}}(A_{n-1})+{\text {Leb}}(B_{n-1}) \le (\tfrac{1}{4}+\tfrac{1}{2a_1}){\text {Leb}}(A_{n-1})\\&\le (\tfrac{1}{2}+\tfrac{1}{a_1}){\text {Leb}}(A_n) =a_0 {\text {Leb}}(A_n), \end{aligned} \end{aligned}$$

completing the proof. $\square$

Lemma 2.9

Let $\varepsilon \in (0,\frac{1}{2}\delta _0)$ be small as in Propositions 2.4 and 2.5. There exist $c_1>0$ and $N\ge 1$ such that

$$\begin{aligned} {\text {Leb}}\left( \bigcup _{i=0}^{N} \{R=n+i\}\right) \ge c_1 {\text {Leb}}(A_{n-1}) \quad \text {for all }n\ge 1. \end{aligned}$$

Proof

Fix $\lambda \in (0,1)$, $L>1$, $0<\delta<\delta _1<\delta _0$ and $N_1\ge 1$ as defined from the outset. Recall that $C_3(3\delta )^\alpha <\frac{1}{2}\delta _0$ and $C_4(L+1)\delta <\delta _0$. Choose $N_2\ge 1$ such that $\lambda ^{N_2}<\varepsilon /\delta _0$ and take $N=N_1+N_2$.

We claim that

• (*) For all $z\in \Lambda$, there exists $i\in \{1,\dots ,N_1\}$ such that $\pi (\phi _{i+N_2} W^u_\varepsilon (z)\cap {\widetilde{{\mathcal {D}}}}_L)\supset {{\mathcal {D}}}_L$.

Fix $z\in \Lambda$. By the definition of $N_1$, there exists $1\le i\le N_1$ such that $d_M(\phi _{-i}p,\phi _{N_2}z)<\delta$. Let $y\in {{\mathcal {D}}}_L$. Then

$$\begin{aligned} d_M(\phi _{-i}y,\phi _{N_2}z){} & {} \le d(\phi _{-i}y,\phi _{-i}p) +d_M(\phi _{-i}p,\phi _{N_2}z)\\{} & {} \le d(y,p)+d_M(\phi _{-i}p,\phi _{N_2}z)<(L+1)\delta . \end{aligned}$$

Using the local product structure and choice of $\delta$, we can define $x\in W^{cs}_{\delta _1}(\phi _{-i}y)\cap W^u_{\delta _1}(\phi _{N_2}z)$. Then $\phi _ix\in W^{cs}_{\delta _1}(y)\subset {\widetilde{{\mathcal {D}}}}_L$ and $g_ix=\pi \phi _ix=y$. Also,

$$\begin{aligned} d(x, \phi _{N_2}z)\le C_4d_M(\phi _{-i}y,\phi _{N_2}z)<C_4(L+1)\delta <\delta _0. \end{aligned}$$

By the definition of $N_2$,

$$\begin{aligned} \phi _ix\in \phi _i W^u_{\delta _0}(\phi _{N_2}z) \subset \phi _{i+N_2} W^u_\varepsilon (z). \end{aligned}$$

Hence we obtain that $y=\pi \phi _ix\in \pi (\phi _{i+N_2}W^u_\varepsilon (z)\cap {\widehat{{\mathcal {D}}}}_L)$ proving (*).

Next, we claim that

• (**) For all $z\in \phi _nA_{n-1}$, $n\ge 1$, there exist $i\in \{0,\dots ,N\}$ and j such that $U_{n+i,j}^1\subset \phi _{-n}W^u_{\delta _0}(z)$.

To prove (**), define $V_\varepsilon =\phi _{-n}W^u_\varepsilon (z)$. By Proposition 2.5(b), $V_\varepsilon \subset A_{n-1}^{(\varepsilon )}$. We now consider two possible cases.

Suppose first that $V_\varepsilon \subset A_{n+i}$ for all $0\le i\le N$. By claim (*), there exists $1\le i\le N=N_1+N_2$ such that

$$\begin{aligned} \pi (\phi _{n+i}V_\varepsilon \cap {\widetilde{{\mathcal {D}}}}_L)=\pi (\phi _iW^u_\varepsilon (z)\cap {\widetilde{{\mathcal {D}}}}_L)\supset {{\mathcal {D}}}_L, \end{aligned}$$

while $V_\varepsilon \subset A_{n+i-1}$ by assumption. This means that $V_\varepsilon \supset U_{n+i,j}^L$ for some j. Hence

$$\begin{aligned} U_{n+i,j}^1\subset U_{n+i,j}^L\subset V_\varepsilon \subset \phi _{-n}W^u_{\delta _0}(z), \end{aligned}$$

and we are done.

In this way, we reduce to the second case where there exists $0\le i\le N$ least such that $V_\varepsilon \not \subset A_{n+i}$. Since i is least, $V_\varepsilon \subset A_{n+i-1}^{(\varepsilon )}$. (The $\varepsilon$ is required in case $i=0$.) By Proposition 2.5(a), $V_\varepsilon \subset \{t_{n+i-1}\le 1\}$. Hence

$$\begin{aligned} \begin{aligned} V_\varepsilon \setminus A_{n+i}&=(V_\varepsilon \cap B_{n+i})\,\cup \,(V_\varepsilon \cap \{R=n+i\})\\&\quad \subset \{t_{n+i-1}\le 1,\,t_{n+i}\ge 1\}\,\cup \, \{R=n+i\} \subset \bigcup _j U_{n+i,j}^2. \end{aligned} \end{aligned}$$

Since $V_\varepsilon \setminus A_{n+i}\ne \emptyset$, this means that there exists j so that $V_\varepsilon$ intersects $U_{n+i,j}^2$. Hence we can choose $a_2\in W^u_\varepsilon (z)\cap \phi _n U_{n+i,j}^2$.

Recall that $\phi _{n+i}U_{n+i,j}^m \subset {\widetilde{{\mathcal {D}}}}_m$ and $g_{n+i}U_{n+i,j}^m = {{\mathcal {D}}}_m$ for $m=1,2$. In particular, $b_2=\phi _ia_2\in {\widetilde{{\mathcal {D}}}}_2$ and $c_2=g_ia_2\in {{\mathcal {D}}}_2$.

Let $c_1\in {{\mathcal {D}}}_1$. Then $d_M(c_1,b_2)\le d_M(c_1,c_2)+d_M(c_2,b_2)<3\delta +\delta _1<4\delta _1$. Hence, using the local product structure and definition of $\delta _1$, we can define $b_1\in W^{cs}_{\delta _0}(c_1)\cap W^u_{\delta _0}(b_2)$ and $a_1=\phi _{-i}b_1$. Note that

$$\begin{aligned} \phi _ia_r=b_r,\qquad \pi b_r=c_r, \quad r=1,2. \end{aligned}$$

Hence

$$\begin{aligned} d(a_1,a_2)\le d(b_1,b_2)\le C_3d(c_1,c_2)^\alpha<C_3(3\delta )^\alpha <\tfrac{1}{2}\delta _0, \end{aligned}$$

and so $d(a_1,z)\le d(a_1,a_2)+d(a_2,z)<\frac{1}{2}\delta _0+\varepsilon <\delta _0$. It follows that $a_1\in W^u_{\delta _0}(z)$ and thereby that $c_1\in g_i(W^u_{\delta _0}(z)\cap \phi _{-i}{\widehat{{\mathcal {D}}}}_1)$. This proves that ${{\mathcal {D}}}_1\subset g_i(W^u_{\delta _0}(z)\cap \phi _{-i}{\widehat{{\mathcal {D}}}}_1)$. Hence $U_{n+i,j}^1\subset g_{-(n+i)}{{\mathcal {D}}}_1\subset \phi _{-n}W^u_{\delta _0}(z)$ verifying claim (**).

We are now in a position to complete the proof of the lemma. Let $n\ge 1$, and let $Z\subset \phi _nA_{n-1}$ be a maximal set of points such that the balls $W^u_{\delta _0/2}(z)$ are disjoint for $z\in Z$. If $x\in \phi _nA_{n-1}$, then $W^u_{\delta _0/2}(x)$ intersects at least one $W^u_{\delta _0/2}(z)$, $z\in Z$, by maximality of the set Z. Hence $\phi _nA_{n-1}\subset \bigcup _{z\in Z}W^u_{\delta _0}(z)$. It follows that

$$\begin{aligned} A_{n-1}\subset \bigcup _{z\in Z}\phi _{-n}W^u_{\delta _0}(z). \end{aligned}$$

Let $z\in Z$ and let $U_z=U_{n+i,j}^1$ be as in claim (**). In particular, $g_{n+i}U_z={{\mathcal {D}}}_1=W^u_\delta (p)$. Also, ${\text {Leb}}(\phi _{n+i}U_z)\le |D\phi _1|E^u|_\infty ^{im}{\text {Leb}}(\phi _nU_z)$ where $m=\dim E^u$. Hence, by (2.2),

$$\begin{aligned} \frac{1}{{\text {Leb}}(\phi _nU_z)}\le |D\phi _1|E^u|_\infty ^{Nm}\frac{1}{{\text {Leb}}(\phi _{n+i}U_z)} \le C_3 |D\phi _1|E^u|_\infty ^{Nm}\frac{1}{{\text {Leb}}(W^u_\delta (p))}. \end{aligned}$$

By (2.1),

$$\begin{aligned} \frac{{\text {Leb}}(\phi _{-n}W^u_{\delta _0}(z))}{{\text {Leb}}(U_z)} \le C_1 \frac{{\text {Leb}}(W^u_{\delta _0}(z))}{{\text {Leb}}(\phi _nU_z)} \le K, \end{aligned}$$

where $K=C_1C_3|D\phi _1|E^u|_\infty ^{Nm}\,\dfrac{\sup _{y\in Y}{\text {Leb}}(W^u_{\delta _0}(y))}{{\text {Leb}}(W^u_\delta (p))}$.

Finally, the sets $U_z$ are connected components of $\bigcup _{0\le i\le N}\{R=n+i\}$ lying in distinct disjoint sets $\phi _{-n}W^u_{\delta _0}(z)$. Hence

$$\begin{aligned} {\text {Leb}}(A_{n-1})\le & {} \sum _{z\in Z} {\text {Leb}}(\phi _{-n}W^u_{\delta _0}(z)) \\\le & {} K \sum _{z\in Z}{\text {Leb}}(U_z) \le K{\text {Leb}}\left( \bigcup _{0\le i\le N}\{R=n+i\}\right) , \end{aligned}$$

as required. $\square$

We can now complete the proof of Theorem 2.1.

Corollary 2.10

${\text {Leb}}(R>n)=O(\gamma ^n)$ for some $\gamma \in (0,1)$.

Proof

By Corollary 2.8 and Lemma 2.9,

$$\begin{aligned} \begin{aligned} {\text {Leb}}(R\ge n)&={\text {Leb}}(A_{n-1})+{\text {Leb}}(B_{n-1})\\&\le (1+a_0){\text {Leb}}(A_{n-1}) \le d_2{\text {Leb}}\left( \bigcup _{i=0}^{N} \{R=n+i\}\right) \end{aligned} \end{aligned}$$

where $d_2=c_1^{-1}(1+a_0)$. It follows that

$$\begin{aligned} \begin{aligned} d_2^{-1}{\text {Leb}}(R\ge n)&\le {\text {Leb}}(R=n)+\dots +{\text {Leb}}(R=n+N)\\&={\text {Leb}}(R\ge n)-{\text {Leb}}(R> n+N). \end{aligned} \end{aligned}$$

Hence

$$\begin{aligned} {\text {Leb}}(R>n+N)\le (1-d_2^{-1}){\text {Leb}}(R\ge n). \end{aligned}$$

In particular, ${\text {Leb}}(R>kN)\le \gamma ^{kN}$ with $\gamma =(1-d_2^{-1})^{1/N}$ and the result follows. $\square$

3 Exponential Decay of Correlations for Flows

In this section, we consider exponential decay of correlations for a class of uniformly hyperbolic skew product flows satisfying a uniform nonintegrability condition, generalising from $C^2$ flows as treated in [8] to $C^{1+\alpha }$ flows. In doing so, we remove the restriction in [6, 9] that unstable manifolds are one-dimensional.

The arguments are a straightforward combination of those in [6, 8]. We follow closely the presentation in [6], with the focus on incorporating the ideas from [8] where required.

Quotienting by stable leaves leads to a class of semiflows considered in Subsection 3.1. The flows are considered in Subsection 3.2.

The current section is completely independent from Sect. 2, so overlaps in notation will not cause any confusion.

3.1 $C^{1+\alpha }$ Uniformly Expanding Semiflows

Fix $\alpha \in (0,1)$. Let $Y\subset {{\mathbb {R}}}^m$ be an open ball⁴ in Euclidean space with Euclidean distance d. We suppose that ${\text {diam}}Y=1$. Let ${\text {Leb}}$ denote Lebesgue measure on Y. Let ${{\mathcal {P}}}$ be a countable partition $\bmod \, 0$ of Y consisting of open sets.

Suppose that $F:\bigcup _{U\in {{\mathcal {P}}}}U\rightarrow Y$ is $C^{1+\alpha }$ on each $U\in {{\mathcal {P}}}$ and maps U diffeomorphically onto Y. Let ${{\mathcal {H}}}=\{h:Y\rightarrow U:U\in {{\mathcal {P}}}\}$ denote the family of inverse branches, and let ${{\mathcal {H}}}_n$ denote the inverse branches for $F^n$. We say that F is a $C^{1+\alpha }$ uniformly expanding map if there exist constants $C_1\ge 1$, $\rho _0\in (0,1)$ such that

• (i) $|Dh|_\infty \le C_1\rho _0^n$ for all $h\in {{\mathcal {H}}}_n$, $n\ge 1$;

• (ii) $|\log |\det Dh|\,|_\alpha \le C_1$ for all $h\in {{\mathcal {H}}}$;

where $|\psi |_\alpha =\sup _{y\ne y'}|\psi (y)-\psi (y')|/d(y,y')^\alpha$. Under these assumptions, it is standard [1] that there exists a unique F-invariant absolutely continuous measure $\mu$. The density $d\mu /d{\text {Leb}}$ is $C^\alpha$, bounded above and below, and $\mu$ is ergodic and mixing.

We consider roof functions $r:\bigcup _{U\in {{\mathcal {P}}}}U\rightarrow {{\mathbb {R}}}^+$ that are $C^1$ on partition elements U with $\inf r>0$. Define the suspension $Y^r=\{(y,u)\in Y\times {{\mathbb {R}}}:0\le u\le r(y)\}/\sim$ where $(y,r(y))\sim (Fy,0)$. The suspension semiflow $F_t:Y^r\rightarrow Y^r$ is given by $F_t(y,u)=(y,u+t)$ computed modulo identifications, with ergodic invariant probability measure $\mu ^r=(\mu \times \textrm{Lebesgue})/{{\bar{r}}}$ where ${{\bar{r}}}=\int _Yr\,d\mu$. We say that $F_t$ is a $C^{1+\alpha }$ uniformly expanding semiflow if F is a $C^{1+\alpha }$ uniformly expanding map and we can choose $C_1$ from condition (i) and $\varepsilon >0$ such that

• (iii) $|D(r\circ h)|_\infty \le C_1$ for all $h\in {{\mathcal {H}}}$;

• (iv) $\sum _{h\in {{\mathcal {H}}}} e^{\varepsilon |r\circ h|_\infty }|\det Dh|_\infty <\infty$.

Let $r_n=\sum _{j=0}^{n-1}r\circ F^j$ and define

$$\begin{aligned} \psi _{h_1,h_2}=r_n\circ h_1-r_n\circ h_2: Y\rightarrow {{\mathbb {R}}}, \end{aligned}$$

for $h_1,h_2\in {{\mathcal {H}}}_n$. We require the following uniform nonintegrability condition [8, Equation (6.6)]:

• (UNI) There exists $E>0$ and $h_1,h_2\in {{\mathcal {H}}}_{n_0}$, for some sufficiently large $n_0\ge 1$, with the following property: There exists a continuous unit vector field $\ell :{{\mathbb {R}}}^m\rightarrow {{\mathbb {R}}}^m$ such that $|D\psi _{h_1,h_2}(y)\cdot \ell (y)|\ge E$ for all $y\in Y$.

(The requirement “sufficiently large” can be made explicit as in [6, Equations (2.1) to (2.3)].) From now on, $n_0$, $h_1$ and $h_2$ are fixed.

Define $F_\alpha (Y^r)$ to consist of $L^\infty$ functions $v:Y^r\rightarrow {{\mathbb {R}}}$ such that $\Vert v\Vert _\alpha =|v|_\infty +|v|_\alpha <\infty$ where

$$\begin{aligned} |v|_\alpha =\sup _{(y,u)\ne (y',u)}\frac{|v(y,u)-v(y',u)|}{d(y,y')^\alpha }. \end{aligned}$$

Define $F_{\alpha ,k}(Y^r)$ to consist of functions with $\Vert v\Vert _{\alpha ,k}=\sum _{j=0}^k \Vert \partial _t^jv\Vert _\alpha <\infty$ where $\partial _t$ denotes differentiation along the semiflow direction.

We can now state the main result in this section. Given $v\in L^1(Y^r)$, $w\in L^\infty (Y^r)$, define the correlation function

$$\begin{aligned} \rho _{v,w}(t)=\int v\,w\circ F_t\,d\mu ^r -\int v\,d\mu ^r \int w\,d\mu ^r. \end{aligned}$$

Theorem 3.1

Suppose that $F_t:Y^r\rightarrow Y^r$ is a $C^{1+\alpha }$ uniformly expanding semiflow satisfying (UNI). Then there exist constants $c,C>0$ such that

$$\begin{aligned} |\rho _{v,w}(t)|\le C e^{-c t}\Vert v\Vert _{\alpha ,1}\Vert w\Vert _{\alpha ,1} \quad \text {for all }t>0\text { and all }v,\,w\in F_{\alpha ,1}(Y^r), \end{aligned}$$

(alternatively, all $v\in F_{\alpha ,2}(Y^r)$, $w\in L^\infty (Y^r)$).

In the remainder of this subsection, we prove Theorem 3.1.

For $s\in {{\mathbb {C}}}$, let $P_s$ denote the (non-normalized) transfer operator

$$\begin{aligned} P_s=\sum _{h\in {{\mathcal {H}}}}A_{s,h}, \qquad A_{s,h}v=e^{-sr\circ h}|\det Dh|\,v\circ h. \end{aligned}$$

For $v:Y\rightarrow {{\mathbb {C}}}$, define $\Vert v\Vert _\alpha =\max \{|v|_\infty ,|v|_\alpha \}$ where $|v|_\alpha =\sup _{y\ne y'}|v(y)-v(y')|/d(y,y')^\alpha$. Let $C^\alpha (Y)$ denote the space of functions $v:Y\rightarrow {{\mathbb {C}}}$ with $\Vert v\Vert _\alpha <\infty$. We introduce the family of equivalent norms

$$\begin{aligned} \Vert v\Vert _b=\max \{|v|_\infty , |v|_\alpha /(1+|b|^\alpha )\}, \quad b\in {{\mathbb {R}}}. \end{aligned}$$

Proposition 3.2

Write $s=\sigma +ib$. There exists $\varepsilon \in (0,1)$ such that the family $s\mapsto P_s$ of operators on $C^\alpha (Y)$ is continuous on $\{\sigma >-\varepsilon \}$. Moreover, $\sup _{|\sigma |<\varepsilon }\Vert P_s\Vert _b<\infty$.

Proof

The first five lines of the proof of [6, Proposition 2.5] should be changed to the following:

Using the inequality $1-t\le -\log t$ valid for $t>0$, we obtain for $a>b>0$ that $a-b=a(1-\frac{b}{a})\le -a\log \frac{b}{a}=a(\log a-\log b).$ Hence $\big ||\det Dh(x)|-|\det Dh(y)|\big |\le |\det Dh|_\infty \big (\log |\det Dh(x)|-\log |\det Dh(y)|\big )$ and so by (ii),

$$\begin{aligned} \big ||\det Dh(x)|-|\det Dh(y)|\big |\le C_1|\det Dh|_\infty \, d(x,y)^\alpha \quad \text {for all }h\in {{\mathcal {H}}}, x,y\in Y. \end{aligned}$$

(3.1)

The proof now proceeds exactly as for [6, Proposition 2.5] (with R, $h'$ and $|x-y|$ changed to r, $\det Dh$ and d(x, y)). $\square$

The unperturbed operator $P_0$ has a simple leading eigenvalue $\lambda _0=1$ with strictly positive $C^\alpha$ eigenfunction $f_0$. By Proposition 3.2, there exists $\varepsilon \in (0,1)$ such that $P_\sigma$ has a continuous family of simple eigenvalues $\lambda _\sigma$ for $|\sigma |<\varepsilon$ with associated $C^\alpha$ eigenfunctions $f_\sigma$. For $s=\sigma +ib$ with $|\sigma |\le \varepsilon$, we define the normalized transfer operators

$$\begin{aligned} L_sv=(\lambda _\sigma f_\sigma )^{-1}P_s(f_\sigma v) =(\lambda _\sigma f_\sigma )^{-1}\sum _{h\in {{\mathcal {H}}}}A_{s,h}(f_\sigma v). \end{aligned}$$

In particular, $L_\sigma 1=1$ and $|L_s|_\infty \le 1$.

Set $C_2=C_1^2/(1-\rho )$, $\rho =\rho _0^\alpha$. Then

($\hbox {ii}_1$):

$|\log |\det Dh|\,|_\alpha \le C_2$ for all $h\in {{\mathcal {H}}}_n$, $n\ge 1$,

($\hbox {iii}_1$):

$|D(r_n\circ h)|_\infty \le C_2$ for all $h\in {{\mathcal {H}}}_n$, $n\ge 1$.

Write

$$\begin{aligned} L_s^nv=\lambda _\sigma ^{-n} f_\sigma ^{-1}\sum _{h\in {{\mathcal {H}}}_n}A_{s,h,n}(f_\sigma v), \quad A_{s,h,n}v=e^{-sr_n\circ h}|\det Dh|v\circ h. \end{aligned}$$

Lemma 3.3

(Lasota-Yorke inequality). There is a constant $C_3>1$ such that

$$\begin{aligned} |L_s^nv|_\alpha \le C_3(1+|b|^\alpha ) |v|_\infty +C_3\rho ^n |v|_\alpha \le C_3(1+|b|^\alpha )\{ |v|_\infty +\rho ^n \Vert v\Vert _b\}, \end{aligned}$$

for all $s=\sigma +ib$, $|\sigma |<\varepsilon$, and all $n\ge 1$, $v\in C^\alpha (Y)$.

Proof

It follows from ($\hbox {ii}_1$) that

$$\begin{aligned} \big ||\det Dh(x)|-|\det Dh(y)|\big |\le C_2|\det Dh|_\infty \, d(x,y)^\alpha \le C_2e^{C_2}|\det Dh(z)|\, d(x,y)^\alpha \end{aligned}$$

for all $h\in {{\mathcal {H}}}_n$, $n\ge 1$, $x,y,z\in Y$. The proof now proceeds exactly as for [6, Lemma 2.7]. $\square$

Corollary 3.4

$\Vert L_s^n\Vert _b\le 2C_3$ for all $s=\sigma +ib$, $|\sigma |<\varepsilon$, and all $n\ge 1$.

Proof

This is unchanged from [6, Corollary 2.8]. $\square$

Given $b\in {{\mathbb {R}}}$, we define the cone

$$\begin{aligned} {{\mathcal {C}}}_b&=\Bigl \{ \;(u,v): u,v\in C^\alpha (Y),\;u>0,\;0\le |v|\le u,\; |\log u|_\alpha \le C_4|b|^\alpha ,\\&\quad |v(x)-v(y)|\le C_4|b|^\alpha u(y)d(x,y)^\alpha \quad \text {for all }x,y\in Y \;\Bigr \}. \end{aligned}$$

(The constant $C_4$ is specified in Lemma 3.8.)

Throughout $B_\delta (y)=\{x\in {{\mathbb {R}}}^m:d(x,y)<\delta \}$.

Lemma 3.5

(Cancellation Lemma). Assume that the (UNI) condition is satisfied (with associated constants $E>0$ and $n_0\ge 1$). Let $h_1,h_2\in {{\mathcal {H}}}_{n_0}$ be the branches from (UNI).

There exists $0<\delta <\Delta =4\pi /E$ such that for all $s=\sigma +ib$, $|\sigma |<\varepsilon$, $|b|\ge 1$, and all $(u,v)\in {{\mathcal {C}}}_b$ we have the following:

For every $y'\in Y$ with $B_{(\delta +\Delta )/|b|}(y')\subset Y$, there exists $y''\in B_{\Delta /|b|}(y')$ such that one of the following inequalities holds on $B_{\delta /|b|}(y'')$:

Case $h_1$::

$|A_{s,h_1,n_0}(f_\sigma v)+ A_{s,h_2,n_0}(f_\sigma v)| \le \tfrac{3}{4} A_{\sigma ,h_1,n_0}(f_\sigma u)+ A_{\sigma ,h_2,n_0}(f_\sigma u)$,

Case $h_2$::

$|A_{s,h_1,n_0}(f_\sigma v)+ A_{s,h_2,n_0}(f_\sigma v)| \le A_{\sigma ,h_1,n_0}(f_\sigma u)+ \tfrac{3}{4} A_{\sigma ,h_2,n_0}(f_\sigma u)$.

Proof

Let $\theta =V-b\psi _{h_1,h_2}$ where $\psi _{h_1,h_2}=r_{n_0}\circ h_1-r_{n_0}\circ h_2$ and $V=\arg (v\circ h_1)-\arg (v\circ h_2)$.

We follow the following steps from [6, Lemma 2.9]:

• (1) Reduce to the situation where $|v(h_my')|> \frac{1}{2} u(h_my')$ for both $m=1$ and $m=2$.

• (2) Establish the estimate $|V(y)-V(y')|\le \pi /6$ for all $y\in B_{(\delta +\Delta )/|b|}(y')$.

• (3) Construct $y''\in B_{\Delta /|b|}(y')$ such that

  $$\begin{aligned} b(\psi _{h_1,h_2}(y'')-\psi _{h_1,h_2}(y'))=\theta (y')-\pi \, \bmod 2\pi . \end{aligned}$$

• (4) Deduce that $|\theta (y)-\pi |\le 2\pi /3$ for all $y\in B_{(\delta +\Delta )/|b|}(y')$.

• (5) Conclude the desired result.

Only step (3) requires any change from the argument in [6, Lemma 2.9]. We provide here the modified argument. Approximate the continuous unit vector field $\ell :{{\mathbb {R}}}^m\rightarrow {{\mathbb {R}}}^m$ in (UNI) by a smooth vector field $\ell :{{\mathbb {R}}}^m\rightarrow {{\mathbb {R}}}^m$ with $|\ell (x)|\le 1$ for all $x\in {{\mathbb {R}}}^m$. By condition ($\hbox {iii}_1$), the approximation can be chosen close enough that

$$\begin{aligned} |D\psi _{h_1,h_2}(y)\cdot \ell (y)|\ge \tfrac{1}{2} E\quad \text {for all }y\in Y. \end{aligned}$$

(3.2)

Let $g:[0,\Delta /|b|]\rightarrow {{\mathbb {R}}}^m$ be the solution to the initial value problem

$$\begin{aligned} \tfrac{dg}{dt}=\ell \circ g, \quad g(0)=y' \end{aligned}$$

and set $y_t=g(t)$. Note that $d(y_t,y')\le \int _0^t|\ell (g(s))|\,ds\le \Delta /|b|$, so $y_t\in B_{\Delta /|b|}(y')$ for all $t\in [0,\Delta /|b|]$. By the mean value theorem applied to $\psi _{h_1,h_2}\circ g:[0,\Delta /|b|]\rightarrow {{\mathbb {R}}}$ and (3.2),

$$\begin{aligned} |\psi _{h_1,h_2}(y_t)-\psi _{h_1,h_2}(y')|\ge t\inf _{s\in [0,\Delta /|b|]}|D\psi _{h_1,h_2}(y_s)\cdot \ell (y_s)| \ge \tfrac{1}{2} Et=(2\pi /\Delta )t \end{aligned}$$

for all $t\in [0,\Delta /|b|]$. It follows that $b(\psi _{h_1,h_2}(y_t)-\psi _{h_1,h_2}(y'))$ fills out an interval around 0 of length at least $2\pi$ as t varies in $[0,\Delta /|b|]$. In particular, we can choose $y''\in B_{\Delta /|b|}(y')$ such that (3) holds. $\square$

Let $\{y_1',\dots ,y_k'\}\subset Y$ be a maximal set of points such that the open balls $B_{(\delta +\Delta )/|b|}(y_i')$ are disjoint and contained in Y.

Let $(u,v)\in {{\mathcal {C}}}_b$. For each $i=1,\dots ,k$, there exists a ball $B_i=B_{\delta /|b|}(y_i'')$ on which the conclusion of Lemma 3.5 holds. Write ${\text {type}}(B_i)=h_m$ if we are in case $h_m$. Let ${{\widehat{B}}}_i=B_{\frac{1}{2}\delta /|b|}(y_i'')$

There exists a universal constant $C>0$ and a $C^1$ function $\omega _i:Y\rightarrow [0,1]$ such that $\omega _i\equiv 1$ on ${{\widehat{B}}}_i$, $\omega _i\equiv 0$ on $Y\setminus B_i$, and $\Vert \omega _i\Vert _{C^1}\le C|b|/\delta$. Define $\omega :Y\rightarrow [0,1]$,

$$\begin{aligned} \omega (y) = {\left\{ \begin{array}{ll} \sum _{{\text {type}}(B_i)=h_m}\omega _i(F^{n_0}y), &{} \quad y\in {\text {range}}{h_m},\,m=1,2 \\ 0 &{} \quad \text {otherwise}. \end{array}\right. } \end{aligned}$$

Note that $\Vert \omega \Vert _{C^1}\le C'|b|$ where $C'=C\delta$ is independent of $(u,v)\in {{\mathcal {C}}}_b$ and $s\in {{\mathbb {C}}}$, and we can assume that $C'>4$. Then $\chi =1-\omega /C':Y\rightarrow [\frac{3}{4},1]$ satisfies $|D\chi |\le |b|$. Moreover, if ${\text {type}}(B_i)=h_m$ then $\chi \equiv \eta$ on ${\text {range}}h_m$ where $\eta =1-1/C'\in (0,1)$.

Corollary 3.6

Let $\delta$, $\Delta$ be as in Lemma 3.5. Let $|b|\ge 1$, $(u,v)\in {{\mathcal {C}}}_b$. Let $\chi =\chi (b,u,v)$ be the $C^1$ function described above (using the branches $h_1,h_2\in {{\mathcal {H}}}_{n_0}$ from (UNI)). Then $|L_s^{n_0}v|\le L_\sigma ^{n_0}(\chi u)$ for all $s=\sigma +ib$, $|\sigma |<\varepsilon$.

Proof

This is immediate from Lemma 3.5 and the definition of $\chi$. $\square$

Define the disjoint union ${{\widehat{B}}}=\bigcup {{\widehat{B}}}_i$.

Proposition 3.7

Let $K>0$. There exists $c_1>0$ such that $\int _{{{\widehat{B}}}}w\,d\mu \ge c_1 \int _Y w\,d\mu$ for all $C^\alpha$ function $w:Y\rightarrow (0,\infty )$ with $|\log w|_\alpha \le K|b|^\alpha$, for all $|b|\ge 16\pi /E$.

Proof

Let $y\in Y$. Since $(\delta +\Delta )/|b|\le 2\Delta /|b|=8\pi /(E|b|)\le \frac{1}{2}$, there exists $z\in Y$ with $B_{(\delta +\Delta )/|b|}(z)\subset Y$ such that $d(z,y)<(\delta +\Delta )/|b|$. By maximality of the set of points $\{y'_1,\dots ,y'_k\}$, there exists $y'_i$ such that $B_{(\delta +\Delta )/|b|}(z)$ intersects $B_{(\delta +\Delta )/|b|}(y'_i)$. Hence $Y\subset \bigcup _{i=1}^k B_i^*$ where $B_i^*= B_{3(\delta +\Delta )/|b|}(y_i')$. Since the density $d\mu /d{\text {Leb}}$ is bounded above and below, there is a constant $c_2>0$ such that $\mu ({{\widehat{B}}}_i)\ge c_2 \mu (B_i^*)$ for each i.

Let $x\in {{\widehat{B}}}_i$, $y\in B_i^*$. Then $d(x,y)\le 4(\delta +\Delta )/|b|$ and so $|w(x)/w(y)|\le e^{K'}$ where $K'= \{4(\delta +\Delta )\}^\alpha K$. It follows that

$$\begin{aligned} \int _{{{\widehat{B}}}_i}w\,d\mu \ge \mu ({{\widehat{B}}}_i)\inf _{{{\widehat{B}}}_i} w \ge c_2 e^{-K'} \mu (B_i^*)\sup _{B_i^*} w \ge c_1 \int _{B_i^*}w\,d\mu , \end{aligned}$$

where $c_1=c_2e^{-K'}$. Since the sets ${{\widehat{B}}}_i\subset Y$ are disjoint,

$$\begin{aligned} \int _{{{\widehat{B}}}} w\,d\mu =\sum _i\int _{{{\widehat{B}}}_i}w\,d\mu \ge c_1 \sum _i\int _{B_i^*}w\,d\mu \ge c_1 \int _Y w\,d\mu \end{aligned}$$

as required. $\square$

Lemma 3.8

(Invariance of cone). There is a constant $C_4$ depending only on $C_1$, $C_2$, $|f_0^{-1}|_\infty$ and $|f_0|_\alpha$ such that the following holds:

For all $(u,v)\in {{\mathcal {C}}}_b$, we have that

$$\begin{aligned} \bigl (\,L_\sigma ^{n_0}(\chi u),\,L_s^{n_0}v\,\bigr )\in {{\mathcal {C}}}_b, \end{aligned}$$

for all $s=\sigma +ib$, $|\sigma |<\varepsilon$, $|b|\ge 1$. (Here, $\chi =\chi (b,u,v)$ is from Corollary 3.6.)

Proof

This is unchanged from [6, Lemma 2.12]. $\square$

Lemma 3.9

($L^2$ contraction). There exist $\varepsilon ,\beta \in (0,1)$ such that

$$\begin{aligned} \int _Y |L_s^{mn_0}v|^2\,d\mu \le \beta ^m|v|_\infty ^2 \end{aligned}$$

for all $m\ge 1$, $s=\sigma +ib$, $|\sigma |<\varepsilon$, $|b|\ge \max \{16\pi /E,1\}$, and all $v\in C^\alpha (Y)$ satisfying $|v|_\alpha \le C_4|b|^\alpha |v|_\infty$.

Proof

Define $u_0\equiv 1, v_0=v/|v|_\infty$ and inductively,

$$\begin{aligned} u_{m+1}=L_\sigma ^{n_0}(\chi _mu_m), \qquad v_{m+1}=L_s^{n_0}(v_m), \end{aligned}$$

where $\chi _m=\chi (b,u_m,v_m)$. It is immediate from the definitions that $(u_0,v_0)\in {{\mathcal {C}}}_b$, and it follows from Lemma 3.8 that $(u_m,v_m)\in {{\mathcal {C}}}_b$ for all m. Hence inductively the $\chi _m$ are well-defined as in Corollary 3.6.

We proceed as in [6, Lemma 2.13] in the following steps.

• (1) It suffices to show that there exists $\beta \in (0,1)$ such that $\int _Y u_{m+1}^2\,d\mu \le \beta \int u_m^2\,d\mu$ for all m.

• (2) Define $w=L_0^{n_0}(u_m^2)$. Then

  $$\begin{aligned} u_{m+1}^2(y)\le {\left\{ \begin{array}{ll} \xi (\sigma )\eta _1 w(y) &{} \quad y\in {{\widehat{B}}}\\ \xi (\sigma ) w(y) &{} \quad y\in Y\setminus {{\widehat{B}}}\end{array}\right. } \end{aligned}$$

  where $\xi (\sigma )$ can be made as close to 1 as desired by shrinking $\varepsilon$. Here, $\eta _1\in (0,1)$ is a constant independent of v, m, s, y.

• (3) The function $w:Y\rightarrow {{\mathbb {R}}}$ satisfies the hypotheses of Proposition 3.7; consequently $\int _{{{\widehat{B}}}}w\,d\mu \ge c_1 \int _{Y{\setminus }{{\widehat{B}}}}w\,d\mu$. This leads to the desired conclusion.

$\square$

Lemma 3.10

($C^\alpha$ contraction). Let $E'=\max \{16\pi /E,2\}$. There exists $\varepsilon \in (0,1)$, $\gamma \in (0,1)$ and $A>0$ such that $\Vert P_s^n\Vert _b\le \gamma ^n$ for all $s=\sigma +ib$, $|\sigma |<\varepsilon$, $|b|\ge E'$, $n\ge A\log |b|$.

Proof

This is unchanged from [6, Proposition 2.14, Corollary 2.15 and Theorem 2.16]. $\square$

Proof of Theorem 3.1

This is identical to [6, Section 2.7]. We note that there is a typo in the statement of [6, Lemma 2.23] where $|b|\le D'$ should be $|b|\ge D'$ (twice). Also, for the second statement of [6, Proposition 2.18] it would be more natural to argue that

$$\begin{aligned} \int _Y e^{\varepsilon r}\,d{\text {Leb}}&=\sum _{h\in {{\mathcal {H}}}}\int _{h(Y)}e^{\varepsilon r}\,d{\text {Leb}}\\&=\sum _{h\in {{\mathcal {H}}}}\int _Y e^{\varepsilon r\circ h}|\det Dh|\,d{\text {Leb}}\le {\text {Leb}}(Y)\sum _{h\in {{\mathcal {H}}}} e^{\varepsilon |r\circ h|_\infty }|\det Dh|_\infty \end{aligned}$$

which is finite by condition (iv). Hence $\int _Y e^{\varepsilon r}\,d\mu <\infty$ by boundedness of $d\mu /d{\text {Leb}}$. 

3.2 $C^{1+\alpha }$ Uniformly Hyperbolic Skew Product Flows

Let $X=Y\times Z$ where Y is an open ball of diameter 1 with Euclidean metric $d_Y$ and $(Z,d_Z)$ is a compact Riemannian manifold. Define the metric $d((y,z),(y',z'))=d_Y(y,y')+d_Z(z,z')$ on X. Let $f(y,z)=(Fy,G(y,z))$ where $F:Y\rightarrow Y$, $G:X\rightarrow Z$ are $C^{1+\alpha }$.

We say that $f:X\rightarrow X$ is a $C^{1+\alpha }$ uniformly hyperbolic skew product if $F:Y\rightarrow Y$ is a $C^{1+\alpha }$ uniformly expanding map satisfying conditions (i) and (ii) as in Sect. 3.1, with absolutely continuous invariant probability measure $\mu$, and moreover

• (v) There exist constants $C>0$, $\gamma _0\in (0,1)$ such that $d(f^n(y,z),f^n(y,z'))\le C\gamma _0^n d(z,z')$ for all $y\in Y$, $z,z'\in Z$.

Let $\pi ^s:X\rightarrow Y$ be the projection $\pi ^s(y,z)=y$. This defines a semiconjugacy between f and F and there is a unique f-invariant ergodic probability measure $\mu _X$ on X such that $\pi ^s_*\mu _X=\mu$.

Suppose that $r:\bigcup _{U\in {{\mathcal {P}}}}U\rightarrow {{\mathbb {R}}}^+$ is $C^1$ on partition elements U with $\inf r>0$. Define $r:X\rightarrow {{\mathbb {R}}}^+$ by setting $r(y,z)=r(y)$. Define the suspension $X^r=\{(x,u)\in X\times {{\mathbb {R}}}:0\le u\le r(x)\}/\sim$ where $(x,r(x))\sim (fx,0)$. The suspension flow $f_t:X^r\rightarrow X^r$ is given by $f_t(x,u)=(x,u+t)$ computed modulo identifications, with ergodic invariant probability measure $\mu _X^r=(\mu _X\times \textrm{Lebesgue})/{{\bar{r}}}$.

We say that $f_t$ is a $C^{1+\alpha }$ uniformly hyperbolic skew product flow provided $f:X\rightarrow X$ is a $C^{1+\alpha }$ uniformly hyperbolic skew product as above, and $r:Y\rightarrow {{\mathbb {R}}}^+$ satisfies conditions (iii) and (iv) as in Sect. 3.1. If $F:Y\rightarrow Y$ and $r:Y\rightarrow {{\mathbb {R}}}^+$ satisfy condition (UNI) from Sect. 3.1, then we say that the skew product flow $f_t$ satisfies (UNI).

Define $F_\alpha (X^r)$ to consist of $L^\infty$ functions $v:X^r\rightarrow {{\mathbb {R}}}$ such that $\Vert v\Vert _\alpha =|v|_\infty +|v|_\alpha <\infty$ where

$$\begin{aligned} |v|_\alpha =\sup _{(y,z,u)\ne (y',z',u)}\frac{|v(y,z,u)-v(y',z',u)|}{d((y,z),(y',z'))^\alpha }. \end{aligned}$$

Define $F_{\alpha ,k}(X^r)$ to consist of functions with $\Vert v\Vert _{\alpha ,k}=\sum _{j=0}^k \Vert \partial _t^jv\Vert _\alpha <\infty$ where $\partial _t$ denotes differentiation along the flow direction.

We can now state the main result in this section. Given $v\in L^1(X^r)$, $w\in L^\infty (X^r)$, define the correlation function

$$\begin{aligned} \rho _{v,w}(t)=\int v\,w\circ f_t\,d\mu _X^r -\int v\,d\mu _X^r \int w\,d\mu _X^r. \end{aligned}$$

Theorem 3.11

Assume that $f_t:X\rightarrow X$ is a $C^{1+\alpha }$ hyperbolic skew product flow satisfying the (UNI) condition. Then there exist constants $c,C>0$ such that

$$\begin{aligned} |\rho _{v,w}(t)|\le C e^{-c t}\Vert v\Vert _{\alpha ,1}\Vert w\Vert _{\alpha ,1}, \end{aligned}$$

for all $t>0$ and all $v,\,w\in F_{\alpha ,1}(X^r)$ (alternatively all $v\in F_{\alpha ,2}(X^r)$, $w\in F_\alpha (X^r)$).

Proof

This is unchanged from [6, Section 4]. $\square$

4 Proof of Theorem 1.2

We return to the situation of Sect. 2, so $\Lambda \subset M$ is a uniformly hyperbolic attractor for a $C^{1+\alpha }$ flow, $\alpha \in (0,1)$, defined on a compact Riemannian manifold. Define the open unstable disk $Y=W^u_\delta (p)$ with discrete return time $R:Y\rightarrow {{\mathbb {Z}}}^+$ and induced map $F=\pi \circ \phi _R:Y\rightarrow Y$ as in Theorem 2.1.

Under smoothness assumptions on holonomies, we verify the conditions on the suspension flow $f_t$ in Sect. 3 and obtain Theorem 1.2 as an easy consequence.

Proposition 4.1

Suppose that the center-stable holonomies are $C^{1+\alpha }$. (In particular, $\pi :{\widehat{{\mathcal {D}}}}\rightarrow {{\mathcal {D}}}$ is $C^{1+\alpha }$.) Then (after shrinking $\delta _0$ in Sect. 2 if necessary) F is a $C^{1+\alpha }$ uniformly expanding map.

Proof

As in Remark 2.2, it is immediate that $F|_U:U\rightarrow Y$ is a $C^{1+\alpha }$ diffeomorphism for all $U\in {{\mathcal {P}}}$. Let $h:Y\rightarrow U$ be an inverse branch with $R|_U=n$, and define $\pi _U=\pi |_{\phi _n(U)}:\phi _n(U)\rightarrow {{\mathcal {D}}}$. Then

$$\begin{aligned} \lambda ^{-1}|v|\le \lambda ^{-n}|v|\le |D\phi _n(x)v|\le |(D\pi _U)^{-1}|_\infty |DF(x)v| \end{aligned}$$

for all $x\in U$, $v\in T_xY$. Hence $|Dh|_\infty \le \rho _0$ where $\rho _0=\lambda \sup _U|(D\pi _U)^{-1}|_\infty$. Shrinking $\delta _0$, we can ensure that $\rho _0<1$. In particular, condition (i) in Sect. 3.1 holds (with $C_1=1$). Condition (ii) is the standard distortion estimate. $\square$

In the remainder of this section, we suppose moreover that the stable holonomies are $C^{1+\alpha }$. Shrink $\delta _0\in (0,1)$ as in Proposition 4.1 and shrink $\delta _1\in (0,\delta _0)$ so that $\phi _t(W^s_{\delta _1}(y))\subset W^s_{\delta _0}(\phi _ty)$ for all $t>0$, $y\in \Lambda$. Recall that ${{\mathcal {D}}}=W^u_{\delta _0}(p)$ and

$$\begin{aligned} {\widehat{{\mathcal {D}}}}=\bigcup _{y\in {{\mathcal {D}}}}W^{cs}_{\delta _0}(y) =\bigcup _{|t|<\delta _0}\phi _t \left( \bigcup _{y\in {{\mathcal {D}}}}W^s_{\delta _0}(y)\right) . \end{aligned}$$

The projection $\pi ^s:\bigcup _{y\in {{\mathcal {D}}}}W^s_{\delta _0}(y)\rightarrow {{\mathcal {D}}}$ given by $\pi ^s|W^s_{\delta _0}(y)\equiv y$ is $C^{1+\alpha }$. Moreover, $\pi =\pi ^s\circ \phi _{r_0}$ where $\phi _{r_0}:{\widehat{{\mathcal {D}}}}\rightarrow \bigcup _{y\in {{\mathcal {D}}}}W^s_{\delta _0}(y)$ and $r_0:{\widehat{{\mathcal {D}}}}\rightarrow (-\delta _0,\delta _0)$ is $C^{1+\alpha }$. Define $r=R+r_0$ on Y. The choice $\delta _0<1$ ensures that $\inf r\ge 1-\delta _0>0$. Define the corresponding semiflow $F_t:Y^r\rightarrow Y^r$.

Proposition 4.2

$F_t:Y^r\rightarrow Y^r$ is a $C^{1+\alpha }$ uniformly expanding semiflow.

Proof

By Proposition 4.1, F is a $C^{1+\alpha }$ uniformly expanding map. In particular, conditions (i) and (ii) are satisfied.

Notice that $F=\pi ^s\circ \phi _r$ where $r=R+r_0:Y\rightarrow {{\mathbb {R}}}^+$ is $C^{1+\alpha }$ on partition elements $U\in {{\mathcal {P}}}$. Since $Dr=Dr_0$ on partition elements, it is immediate that $\sup _{h\in {{\mathcal {H}}}}|D(r\circ h)|_\infty \le |D r_0|_\infty \sup _{h\in {{\mathcal {H}}}}|Dh|_\infty \le \rho _0|Dr_0|_\infty <\infty$ verifying condition (iii) on r. Recall that ${\text {Leb}}(R>n)=O(\gamma ^n)$ for some $\gamma \in (0,1)$, so we can choose $\varepsilon >0$ such that $\int _Y e^{\varepsilon R}\,d{\text {Leb}}<\infty$. Condition (ii) ensures that $|\det Dh|_\infty \le ({\text {Leb}}Y)^{-1}e^{C_1}{\text {Leb}}({\text {range}}h)$ for all $h\in {{\mathcal {H}}}$. Hence $\sum _{h\in {{\mathcal {H}}}}e^{\varepsilon |r\circ h|_\infty }|\det Dh|_\infty \ll \sum _{h\in {{\mathcal {H}}}}e^{\varepsilon |R\circ h|_\infty }{\text {Leb}}({\text {range}}h)=\int _Y e^{\varepsilon R}\,d{\text {Leb}}<\infty$ verifying condition (iv) on r. $\square$

We now make a $C^{1+\alpha }$ change of coordinates so that ${\widehat{{\mathcal {D}}}}$ is identified with ${{\mathcal {D}}}\times W_{\delta _0}^{s}(p)\times (-\delta _0,\delta _0)$ where $\{y\}\times W_{\delta _0}^{s}(p)$ is identified with $W_{\delta _0}^{s}(y)$ for all $y\in {{\mathcal {D}}}$ and $(-\delta _0,\delta _0)$ is the flow direction. Let $X=Y\times Z$ where $Z= W_{\delta _0}^{s}(p)$ and define $r:X\rightarrow (0,\infty )$ by $r(y,z)=r(y)$. Also, define $f=\phi _r:X\rightarrow X$ and the corresponding suspension flow $f_t:X^r\rightarrow X^r$

Proposition 4.3

$f_t:X^r\rightarrow X^r$ is a $C^{1+\alpha }$ uniformly hyperbolic skew product flow.

Proof

Note that $\pi ^s(X)=Y$ and $\pi ^s(y,z)=y$. Also, $f(y,z)=(Fy,G(y,z))$ where $G:X\rightarrow Z$ is $C^{1+\alpha }$. Since Z corresponds to the exponential contracting stable foliation, condition (v) in Sect. 3.2 is satisfied. Hence $f:X\rightarrow X$ is a $C^{1+\alpha }$ uniformly hyperbolic skew product and the corresponding suspension flow $f_t:X^r\rightarrow X^r$ is a $C^{1+\alpha }$ uniformly hyperbolic skew product flow. $\square$

Next we recall the standard argument that joint nonintegrability implies (UNI) in the current situation. (Similar arguments are given for instance in [7, Section 3] and [21, Section 5.3].)

Joint nonintegrability is defined in terms of the temporal distortion function. To define this intrinsically (independently of the inducing scheme) we have to introduce the first return time $\tau :X\rightarrow {{\mathbb {R}}}^+$ and the Poincaré map $g:X\rightarrow X$ given by

$$\begin{aligned} \tau (x)=\inf \{t>0:\phi _t(x)\in X\}, \qquad g(x)=\phi _{\tau (x)}(x). \end{aligned}$$

Note that $\tau$ is constant along stable leaves by the choice of X.

For $x_1,x_2\in X$, define the local product $[x_1,x_2]$ to be the unique intersection point of $W^u(x_1)\cap W^s(x_2)$. The temporal distortion function D is defined to be

$$\begin{aligned} D(x_1,x_2) =\sum _{j=-\infty }^\infty \big \{ \tau (g^jx_1) -\tau (g^j[x_1,x_2]) -\tau (g^j[x_2,x_1])+\tau (g^jx_2)\big \} \end{aligned}$$

at points $x_1,x_2\in X$. The stable and unstable bundles are jointly integrable if and only if $D\equiv 0$.

Lemma 4.4

Joint nonintegrability of the stable and unstable bundles implies (UNI).

Proof

For points $x,x'\in X$ with $x'\in W^u(x)$, we define

$$\begin{aligned} D_0(x,x')=\sum _{j=1}^\infty \big \{\tau (g^{-j}x)-\tau (g^{-j}x')\big \}. \end{aligned}$$

Since $\tau$ is constant along stable leaves,

$$\begin{aligned} D(x_1,x_2)&=\sum _{j=1}^\infty \big \{ \tau (g^{-j}x_1) -\tau (g^{-j}[x_1,x_2]) -\tau (g^{-j}[x_2,x_1])+\tau (g^{-j}x_2)\big \}\\&= D_0(x_1,[x_1,x_2])+D_0(x_2,[x_2,x_1]). \end{aligned}$$

Next, we find a more convenient expression for $D_0$ in terms of r and f. Note that for any $x\in X$, there exists $N(x)\in {{\mathbb {Z}}}^+$ (the number of returns to X up to time r(x)) such that

$$\begin{aligned} r(x)=\sum _{\ell =0}^{N(x)-1}\tau (g^\ell x), \qquad f(x)=g^{N(x)}x. \end{aligned}$$

Corresponding to the partition ${{\mathcal {P}}}$ of Y, we define the collection ${\widetilde{{\mathcal {P}}}}=\{{{\bar{U}}}\times {{\bar{Z}}}:U\in {{\mathcal {P}}}\}$ of closed subsets of X. Suppose that $x,x'\in V_0$, $V_0\in {\widetilde{{\mathcal {P}}}}$, with $x'\in W^u(x)$. The induced map $f:X\rightarrow X$ need not be invertible since it is not the first return to X. However, we may construct suitable inverse branches $z_j$, $z_j'$ of x, $x'$ as follows. Set $z_0=x$, $z_0'=x'$. Since f is transitive and continuous on closures of partition elements, there exists $V_1\in {\widetilde{{\mathcal {P}}}}$ and $z_1\in V_1$ such that $fz_1=z_0$. Since F is full-branch, $f(W^u(z_1)\cap V_1)\supset W^u(z_0)$, so there exists $z_1'\in W^u(z_1)\cap V_1$ such that $fz_1'=z_0'$. Inductively, we obtain $V_n\in {\widetilde{{\mathcal {P}}}}$ and $z_j,z_j'\in V_n$ with $z_j'\in W^u(z_j)$ such that $fz_j=z_{j-1}$ and $fz_j'=z_{j-1}'$.

By construction, $z_{j-1}=fz_j=g^{N(z_j)}z_j$. Hence $z_j=g^{-(N(z_1)+\dots +N(z_j))}x$ and

$$\begin{aligned} r(z_j) =\sum _{\ell =0}^{N(z_j)-1}\tau (g^\ell g^{-(N(z_1)+\dots +N(z_j))}x) =\sum _{\ell =N(z_1)+\dots +N(z_{j-1})+1}^{N(z_1)+\dots +N(z_j)}\tau (g^{-\ell }x). \end{aligned}$$

A similar expression holds for $r(z_j')$. Hence

$$\begin{aligned} D_0(x,x')= \sum _{j=1}^\infty \big \{ r(z_j) -r(z_j')\big \}. \end{aligned}$$

We are now in a position to complete the proof of the lemma, showing that if (UNI) fails, then $D\equiv 0$. To do this, we make use of [8, Proposition 7.4] (specifically the equivalence of their conditions 1 and 3). Namely, the failure of the (UNI) condition in Sect. 3.1 means that we can write $r=\xi \circ F-\xi +\zeta$ on Y where $\xi :Y\rightarrow {{\mathbb {R}}}$ is continuous (even $C^1$) and $\zeta$ is constant on partition elements $U\in {{\mathcal {P}}}$. Extending $\xi$ and $\zeta$ trivially to $X=Y\times Z$, we obtain that $r=\xi \circ f-\xi +\zeta$ on X where $\xi :X\rightarrow {{\mathbb {R}}}$ is continuous and constant on stable leaves, and $\zeta$ is constant on elements $V\in {\widetilde{{\mathcal {P}}}}$. In particular,

$$\begin{aligned} \sum _{j=1}^n r(z_j)=\sum _{j=1}^n \big \{\xi (z_{j-1}-\xi (z_j)+\zeta (z_j)\big \} =\xi (x)-\xi (z_n)+\sum _{j=1}^n \zeta (z_j). \end{aligned}$$

For $x,x'\in V_0$, $V_0\in {\widetilde{{\mathcal {P}}}}$, with $x'\in W^u(x)$, it follows that

$$\begin{aligned} \sum _{j=1}^n \big \{ r(z_j) -r(z_j')\} =\xi (x)-\xi (x')- \xi (z_n) +\xi (z_n'). \end{aligned}$$

Taking the limit as $n\rightarrow \infty$, we obtain that $D_0(x,x')=\xi (x)-\xi (x')$. Hence $D(x_1,x_2)=\xi (x_1)-\xi ([x_1,x_2])-\xi ([x_2,x_1])+\xi (x_2)$. Since $\xi$ is constant on stable leaves, $D(x_1,x_2){=0}$ as required. $\square$

Proof of Theorem 1.2

By Proposition 4.3 and Lemma 4.4, $f_t$ is a $C^{1+\alpha }$ uniformly hyperbolic flow satisfying (UNI). The result for $C^{1+\alpha }$ observables follows from Theorem 3.11. As in [18], the result follows from a standard interpolation argument (see also [6, Corollary 2.3]).