Julia Sets of Hyperbolic Rational Maps have Positive Fourier Dimension

Abstract

Let $f:\widehat{{\mathbb {C}}}\rightarrow \widehat{{\mathbb {C}}}$ be a hyperbolic rational map of degree $d \ge 2$, and let $J \subset {\mathbb {C}}$ be its Julia set. We prove that J always has positive Fourier dimension. The case where J is included in a circle follows from a recent work of Sahlsten and Stevens (Fourier transform and expanding maps on Cantor sets preprint. arXiv:2009.01703, 2020). In the case where J is not included in a circle, we prove that a large family of probability measures supported on J exhibit polynomial Fourier decay: our result applies in particular to the measure of maximal entropy and to the conformal measure.

Appendices

Appendix A Large Deviations

The goal of this section is to prove the large deviation Theorem 2.8, by using properties of the pressure.

The link between the spectral radius of ${\mathcal {L}}_\varphi$ and the pressure given by the Perron-Frobenius-Ruelle theorem allows us to get the following useful formula. We extract the first one from [Ru78], Theorem 7.20 and remark 7.28, and the second one from [Ru89], lemma 4.5.

Proposition A.1

 

$$\begin{aligned} P(\varphi )= & {} \lim _{n \rightarrow \infty } \frac{1}{n} \log \sum _{f^n(x)=x} e^{S_n \varphi (x)}\\ P(\varphi )= & {} \lim _{n \rightarrow \infty } \frac{1}{n} \max _{b \in {\mathcal {A}}} \sup _{x \in P_b} \log \underset{{\mathbf {a}} \rightsquigarrow b}{\sum _{{{\textbf {a}}} \in {\mathcal {W}}_{n+1}} } e^{S_n \varphi ( g_{{{\textbf {a}}}}(x) )} \end{aligned}$$

We begin by proving another avatar of those spectral radius formulas (which is nothing new).

Lemma A.2

Choose any $x_{{{\textbf {a}}}}$ in each of the $P_{{{\textbf {a}}}}$, ${{\textbf {a}}} \in {\mathcal {W}}_{n}$, $\forall n$. Then

$$\begin{aligned} P(\varphi ) = \lim _n \frac{1}{n} \log \sum _{{{\textbf {a}}} \in {\mathcal {W}}_{n+1}} e^{S_n \varphi (x_{{\textbf {a}}})} . \end{aligned}$$

Proof

Since $P_{{{\textbf {a}}}}$ is compact, and by continuity, for every n there exists $b^{(n)} \in {\mathcal {A}}$ and $y_{b^{(n)}}^{(n)} \in P_{b^{(n)}}$ such that

$$\begin{aligned} \max _{b \in {\mathcal {A}}} \sup _{x \in P_b} \log \underset{{\mathbf {a}} \rightsquigarrow b}{\sum _{{{\textbf {a}}} \in {\mathcal {W}}_{n+1}} } e^{S_n \varphi ( g_{{{\textbf {a}}}}(x) )} = \log \underset{{\mathbf {a}} \rightsquigarrow b^{(n)}}{\sum _{{{\textbf {a}}} \in {\mathcal {W}}_{n+1}} } e^{S_n \varphi ( g_{{{\textbf {a}}}}(y_{b^{(n)}}^{(n)}) )}. \end{aligned}$$

Define $y_{{{\textbf {a}}}} := g_{{{\textbf {a}}}}(y_{b^{(n)}}^{(n)}) \in P_{{{\textbf {a}}}}$ for clarity. The dependence on n is not lost since it is contained in the length of the word. First of all, since $\varphi$ has exponentially vanishing variations, there exists a constant $C_1>0$ such that

$$\begin{aligned} \forall x,y \in P_{{{\textbf {a}}}}, \ | S_n \varphi (x) - S_n \varphi (y) |\le C_1 . \end{aligned}$$

Now we want to relate the sums with the $x_{\mathbf {a}}$’s and the $y_{{\mathbf {a}}}$’s, but the indices are different. To do it properly, we are going to use the fact that f is topologically mixing: there exists some $N \in {\mathbb {N}}$ such that the matrix $M^N$ has all its entries positive. In particular, it means that

$$\begin{aligned} \forall b \in {\mathcal {A}}, \ \forall {{\textbf {a}}} \in {\mathcal {W}}_{n+1}, \ \exists {{\textbf {c}}} \in {\mathcal {W}}_{N}, \ {\textbf {ac}}b \in {\mathcal {W}}_{n+N+1} . \end{aligned}$$

The point is that we are sure that the word is admissible.

For a given ${{\textbf {a}}} \in {\mathcal {W}}_{n+1}$, there exists a ${{\textbf {c}}} \in {\mathcal {W}}_{N}$ such that ${\textbf {ac}}b^{(n+N+1)} \in {\mathcal {W}}_{n+N+1}$, and so, using the fact that $e^{S_n \varphi } \ge 0$, we get:

$$\begin{aligned} e^{S_n \varphi (x_{{{\textbf {a}}}})} \le \underset{ {\textbf {ac}}b^{(n+N+1)} \in {\mathcal {W}}_{n+N+1}}{\sum _{{{\textbf {c}}} \in {\mathcal {W}}_{N} }} e^{C_1} e^{S_n \varphi (y_{{\textbf {ac}}b^{(n+N+1)}})} . \end{aligned}$$

Then, since $S_{n}(\varphi ) \le S_{n+N}(\varphi ) +N \Vert \varphi \Vert _{\infty ,J}$, we have:

$$\begin{aligned} e^{S_n \varphi (x_{{{\textbf {a}}}})} \le \underset{ {\textbf {ac}}b^{(n+N+1)} \in {\mathcal {W}}_{n+N+1}}{\sum _{{{\textbf {c}}} \in {\mathcal {W}}_{N} }} e^{C_2} e^{S_{n+N}(\varphi )(y_{{\textbf {ac}}b^{(n+N+1)}})} . \end{aligned}$$

Hence

$$\begin{aligned} \log \left( {\sum _{{{\textbf {a}}} \in {\mathcal {W}}_{n+1}} } e^{S_n \varphi ( x_{{{\textbf {a}}}} )} \right)\le & {} \log \left( \sum _{{{\textbf {a}}} \in {\mathcal {W}}_{n+1}} \underset{ {\textbf {ac}}b^{(n+N+1)} \in {\mathcal {W}}_{n+N+1}}{\sum _{{{\textbf {c}}} \in {\mathcal {W}}_{N} }} e^{C_2} e^{S_{n+N} \varphi (y_{{\textbf {ac}}b^{(n+N+1)}})} \right) \\= & {} C_2 + \log \left( \underset{{\mathbf {d}} \rightsquigarrow b^{(n+N+1)}}{\sum _{{{\textbf {d}}} \in {\mathcal {W}}_{n+N+1} }} e^{S_n \varphi (y_{{{\textbf {d}}}})} \right) , \end{aligned}$$

and so

$$\begin{aligned} \limsup _{n \rightarrow \infty } \frac{1}{n} \log \left( {\sum _{{{\textbf {a}}} \in {\mathcal {W}}_{n+1}} } e^{S_n \varphi ( x_{{{\textbf {a}}}} )} \right) \le P(\varphi ) . \end{aligned}$$

The other inequality is easier, we have

$$\begin{aligned} \log \left( \underset{ {\mathbf {a}} \rightsquigarrow b^{(n)}}{\sum _{{{\textbf {a}}} \in {\mathcal {W}}_{n+1}} } e^{S_n \varphi ( y_{{{\textbf {a}}}} )} \right) \le C_1 + \log \left( \sum _{{{\textbf {a}}} \in {\mathcal {W}}_{n+1}} e^{S_n \varphi (x_{{{\textbf {a}}}})} \right) , \end{aligned}$$

which gives us

$$\begin{aligned} P(\varphi ) \le \liminf \frac{1}{n} \log \left( \sum _{{{\textbf {a}}} \in {\mathcal {W}}_{n+1}} e^{S_n \varphi (x_{{{\textbf {a}}}})} \right) . \end{aligned}$$

$\square$

Another useful formula is the computation of the differential of the pressure.

Theorem A.3

The map $P : C^1(U,{\mathbb {R}}) \rightarrow {\mathbb {R}}$ is differentiable. If $\varphi \in C^1(U,{\mathbb {R}})$ is a normalized potential, then we have:

$$\begin{aligned} \forall \psi \in C^1(U,{\mathbb {R}}), \ (dP)_{\varphi }(\psi ) = \int _J \psi d\mu _{\varphi } . \end{aligned}$$

Proof

This is the corollary 5.2 in [Ru89]. Loosely, the argument goes as follows.

The differentiability is essentially a consequence of the fact that $e^{P(\psi )}$ is an isolated eigenvalue of ${\mathcal {L}}_\psi$. To compute the differential, consider $v_t \in C^1$ the normalized eigenfunction for ${\mathcal {L}}_{\varphi + t \psi }$ such that $v_0=1$. We have, for small t:

$$\begin{aligned} {\mathcal {L}}_{\varphi + t \psi } v_t = e^{P(\varphi + t \psi )} v_t \end{aligned}$$

Hence,

$$\begin{aligned} {\mathcal {L}}_{\varphi + t \psi } (\psi v_t ) + {\mathcal {L}}_{\varphi + t \psi } (\partial _t v_t) = v_t e^{P(\varphi + t \psi )} \frac{d}{dt} P(\varphi + t \psi ) + e^{P(\varphi + t \psi )} \partial _t v_t \end{aligned}$$

Taking $t=0$ and integrating against $\mu _\varphi$ gives

$$\begin{aligned} (d P)_{\varphi }(\psi ) = \int _J {\mathcal {L}}_{\varphi }(\psi ) d\mu _\varphi = \int _J \psi d\mu _\varphi . \end{aligned}$$

$\square$

Now, we are ready to prove Theorem 2.8. The proof is adapted from [JS16], subsection 4.

Proof

Let $\varphi$ be a normalized potential, and let $\psi$ be another $C^1$ potential. Let $\varepsilon > 0$. Let $j(t) := P\left( (\psi - \int \psi d\mu _\varphi - \varepsilon )t + \varphi \right)$. We know by Theorem A.3 that $j'(0) = -\varepsilon < 0$. Hence, there exists $t_0> 0$ such that $P( (\psi - \int \psi d\mu _\varphi - \varepsilon )t_0 + \varphi ) < 0$.

Define $2 \delta _0 := -P\left( (\psi - \int \psi d\mu _\varphi - \varepsilon )t_0 + \varphi \right)$. We then have

$$\begin{aligned} \mu _\varphi \left( \left\{ x \in J \ , \ \frac{1}{n} S_n \psi (x) - \int _J \psi d\mu _{\varphi } \ge \varepsilon \right\} \right) \le \sum _{{{\textbf {a}}} \in C_{n+1}} \mu _\varphi (P_{\mathbf {a}}) , \end{aligned}$$

where $C_{n+1} := \{ {\mathbf {a}} \in {\mathcal {W}}_{n+1} \ | \ \exists x \in P_{{\mathbf {a}}} , \ S_n \psi (x)/n - \int \psi d\mu _{\varphi } \ge \varepsilon \}$. For each ${\mathbf {a}}$ in some $C_{n+1}$, choose $x_{{\mathbf {a}} } \in P_{{\mathbf {a}} }$ such that $S_n \psi (x_{{\mathbf {a}} })/n - \int \psi d\mu _{\varphi } \ge \varepsilon$. For the other ${\mathbf {a}}$, choose $x_{{\mathbf {a}}} \in P_{{{\textbf {a}}}}$ randomly.

Now, since $\mu _\varphi$ is a Gibbs measure, there exists $C_0>0$ such that:

$$\begin{aligned} \sum _{{{\textbf {a}}} \in C_{n+1}} \mu _\varphi (P_{\mathbf {a}})\le & {} C_0 \sum _{{{\textbf {a}}} \in C_{n+1}} \exp (S_n \varphi (x_{{\textbf {a}}})) \\\le & {} C_0 \sum _{{{\textbf {a}}} \in C_{n+1}} \exp \left( {S_n\left( \left( \psi -\int \psi d\mu _\varphi - \varepsilon \right) t_0 + \varphi \right) (x_{{\textbf {a}}} )} \right) \\\le & {} C_0 \sum _{{{\textbf {a}}} \in {\mathcal {W}}_{n+1}} \exp \left( {S_n\left( \left( \psi -\int \psi d\mu _\varphi - \varepsilon \right) t_0 + \varphi \right) (x_{{\textbf {a}}} )} \right) . \end{aligned}$$

Then, by the lemma A.2, we can write for $n \ge n_0$ large enough:

$$\begin{aligned}&C_0 \sum _{{{\textbf {a}}} \in {\mathcal {W}}_{n+1}} \exp \left( {S_n\left( \left( \psi -\int \psi d\mu _\varphi - \varepsilon \right) t_0 + \varphi \right) (x_{{\textbf {a}}} )} \right) \\&\quad \le C e^{n \delta _0} e^{n P( (\psi - \int \psi d\mu _\varphi - \varepsilon )t_0 + \varphi )} \le C e^{- \delta _0 n} , \end{aligned}$$

and so

$$\begin{aligned} \mu _\varphi \left( \left\{ x \in J \ , \ \frac{1}{n} S_n \psi (x) - \int _X \psi d\mu _{\varphi } \ge \varepsilon \right\} \right) \le Ce^{- n \delta _0} . \end{aligned}$$

The symmetric case is done by replacing $\psi$ by $-\psi$, and combining the two gives us the desired bound. $\square$

Appendix B Uniform Spectral Estimate for a Family of Twisted Transfer Operator

Here, we will show how to prove Theorem 6.4. It is a generalization of Theorem 2.5 in [OW17]: we will explain what we need to change in the original paper for the theorem to hold more generally.

Proving that a complex transfer operator is eventually contracting is linked to analytic extensions results for dynamical zeta functions, and is often referred to as a spectral gap. Such results are of great interest to study, for example, periodic orbit distribution in hyperbolic dynamical systems (see for example the chapter 5 and 6 in [PP90]), or asymptotics for dynamically defined quantities (as in [OW17] or [PU17]). One of the first result of this kind can be found in a work of Dolgopyat [Do98], in which he used a method that has been broadly extended since. We can find various versions of Dolgopyat’s method in papers of Naud [Na05], Stoyanov [St11], Petkov [PS16], Oh-Winter [OW17], Li [Li18b], and Sharp-Stylianou [ShSt20], to only name a few.

In this annex, we will outline the argument of Dolgopyat’s method as explained in [OW17] adapted to our general setting. We need three ingredients to make the method work: the NLI (non local integrability), the NCP (another non concentration property), and a doubling property.

Definition B.1

Define $\tau (x) := \log |f'(x)| \in {\mathbb {R}}$ and $\theta (x) := \arg f'(x) \in {\mathbb {R}}/2 \pi {\mathbb {Z}}.$ The transfer operator in Theorem 6.4 acts on $C^1_b({\mathfrak {U}},{\mathbb {C}})$ and may be rewritten in the form

$$\begin{aligned} {\mathcal {L}}_{\varphi ,it,l} = {\mathcal {L}}_{\varphi - i t \mathcal {\tau } - i l \mathcal {\theta }}. \end{aligned}$$

For some normalized $\varphi \in C^1(U,{\mathbb {R}})$, $l \in {\mathbb {Z}}$ and $t \in {\mathbb {R}}$.

With those notations, Theorem 6.4 can be rewritten as follows.

Theorem B.1

Suppose that J is not included in a circle. For any $\varepsilon >0$, there exists $C>0$, $\rho <1$ such that for any $n \ge 1$ and any $t\in {\mathbb {R}}$, $l \in {\mathbb {Z}}$ such that $|t|+|l| > 1$,

$$\begin{aligned} \Vert {\mathcal {L}}_{\varphi - i(t \tau + l \theta )}^n \Vert _{C^1_b({\mathfrak {U}},{\mathbb {C}})} \le C (|l|+|t|)^{1+\varepsilon } \rho ^n \end{aligned}$$

Now we may recall the three main technical ingredients.

Theorem B.2

(NLI, [OW17] section 3). The function $\tau$ satisfies the NLI property if there exists $a_0 \in {\mathcal {A}}$, $x_1 \in P_{a_0}$ , $N \in {\mathbb {N}}$, admissible words ${\mathbf {a}}, {\mathbf {b}} \in {\mathcal {W}}_{N+1}$ with $a_0\rightsquigarrow {\mathbf {a}},{\mathbf {b}}$, and an open neighborhood $U_0$ of $x_1$ such that for any $n \ge N$, the map

$$\begin{aligned} ({\tilde{\tau }}, {\tilde{\theta }}) := (S_n \tau \circ g_{\mathbf {a}} - S_n \tau \circ g_{\mathbf {b}}, S_n\theta \circ g_{\mathbf {a}} - S_n\theta \circ g_{\mathbf {b}}) : U_0 \rightarrow {\mathbb {R}} \times {\mathbb {R}}/2 \pi {\mathbb {Z}} \end{aligned}$$

is a local diffeomorphism.

Remark B.1

Remark 4.7 in [SS20] and Proposition 3.8 in [OW17] points out the fact that the NLI is a consequence of our non-linear setting, which itself comes from the fact that we supposed that our Julia set is different from a circle.

Theorem B.3

(NCP, [OW17] section 4). For each $n \in {\mathbb {N}}$, for any ${\mathbf {a}} \in {\mathcal {W}}_{n+1}$, there exists $0< \delta < 1$ such that, for all $x \in P_{\mathbf {a}}$, all $w \in {\mathbb {C}}$ of unit length, and all $\varepsilon \in (0, 1)$,

$$\begin{aligned} B(x,\varepsilon ) \cap \{ y \in P_{\mathbf {a}}, \ |\langle y - x, w \rangle | > \delta \varepsilon \} \ne \emptyset \end{aligned}$$

where $\langle a + bi, c + di \rangle = ac + bd$ for $a, b, c, d \in {\mathbb {R}}$.

Remark B.2

The NCP is a consequence of the fractal behavior of our Julia set. This time, if J is included in any smooth set, the NCP fails. But in our case, this is equivalent to being included in a circle, see [ES11]. Notice that this non concentration property has nothing to do with our previous non concentration hypothesis.

Theorem B.4

(Doubling). Let, for $a \in {\mathcal {A}}$, $\mu _a$ be the equilibrium measure $\mu _\varphi$ restricted to $P_a$. Then each $\mu _a$ is doubling, that is:

$$\begin{aligned} \exists C>0, \ \forall x \in P_a, \ \forall r<1, \ \mu _a(B(x,2r)) \le C \mu _a(B(x,r)). \end{aligned}$$

Proof

It follows from Theorem A.2 in [PW97] that $\mu _\varphi$ is doubling: the proof uses the conformality of the dynamics. To prove that $\mu _a := \mu _{| P_a}$ is still doubling, which is not clear a priori, we follow Proposition 4.5 in [OW17] and prove that there exists $c>0$ such that for any $a \in {\mathcal {A}}$, for any $x \in P_a$, and for any $r>0$ small enough,

$$\begin{aligned} \frac{\mu _\varphi (B(x,r) \cap P_a)}{\mu _\varphi (B(x,r))} > c . \end{aligned}$$

For this we use a Moran cover ${\mathcal {P}}_r$ associated to our Markov partition, see the proof of Proposition 2.7 for a definition. Recall that any element $P \in {\mathcal {P}}_r$ have diameter strictly less than r, and recall that there exists a constant $M>0$ independent of x and r such that we can cover the ball B(x, r) with M elements of ${\mathcal {P}}_r$. Moreover, lemma 2.2 in [WW17] allows us to do so using elements $P\in {\mathcal {P}}_r$ of the form $P_{\mathbf {a}}$ for ${\mathbf {a}}$ in some ${\mathcal {W}}_n$, $N_0 \le n \le N_0 + L$ for some $N_0(x,r)$ and some constant L (independent of x and r). We can then conclude as follows. Let $P^{(1)} , \dots P^{(M)} \in {\mathcal {P}}_r$ that covers B(x, r). There exists i such that $P^{(i)} \subset P_a$. Hence, by the Gibbs property of $\mu _\varphi$:

$$\begin{aligned} \frac{\mu _\varphi (B(x,r) \cap P_a)}{\mu _\varphi (B(x,r))} \ge \frac{\mu _\varphi (P^{(i)})}{ {\sum _{j=1}^M} \mu _\varphi (P^{(j)})} \ge M^{-1} C_0^{-2} e^{-(L+2)\Vert \varphi \Vert _\infty }. \end{aligned}$$

$\square$

Remark B.3

The doubling property (or Federer property) is a regularity assumption made on the measure that is central for the execution of this version of Dolgopyat’s method. It allows us to control integrals over J by integrals over smaller pieces of J, provided some regularity assumption on the integrand.

Now we will outline the argument of Dolgopyat’s method as used in [OW17]. It can be decomposed into four main steps.

Step 1 We reduce Theorem B.1 to a $L^2(\mu )$ estimate.

We need to define a modified $C^1$ norm. Denote by $\Vert \cdot \Vert _r$ a new norm, defined by

$$\begin{aligned} \Vert h\Vert _r := \left\{ \begin{array}{rcl} \Vert h\Vert _{\infty ,{\mathfrak {U}}} + \frac{\Vert \nabla h\Vert _{\infty ,{\mathfrak {U}}}}{r} \ \text {if} \ r \ge 1 \\ \Vert h\Vert _{\infty ,{\mathfrak {U}}} + \Vert \nabla h \Vert _{\infty ,{\mathfrak {U}}} \ \text {if} \ r < 1 \end{array} \right. \end{aligned}$$

Moreover, we do a slight abuse of notation and write $\mu$ for $\sum _{a \in {\mathcal {A}}} \mu _a$, seen as a measure on ${\mathfrak {U}}$. This measure is supported on J, seen as the set $\bigsqcup _a P_a \subset \bigsqcup _a U_a = {\mathfrak {U}}$. The first step is to show that Theorem B.1 reduces to the following claim.

Theorem B.5

Suppose that the Julia set of f is not contained in a circle. Then there exists $C>0$ and $\rho \in (0,1)$ such that for any $h \in C^1_b({\mathfrak {U}},{\mathbb {C}})$ and any $n \in {\mathbb {N}}$,

$$\begin{aligned} ||{\mathcal {L}}_{\varphi -i(t \tau + l \theta )}^n h||_{L^2(\mu )} \le C \rho ^n \Vert h \Vert _{|t|+|l|} \end{aligned}$$

for all $t \in {\mathbb {R}}$ and $\ell \in {\mathbb {Z}}$ with $|t| + |l| \ge 1$.

A clear account for this reduction may be found in [Na05], section 5. This step holds in great generality without any major difficulty. Intuitively, Theorem B.1 follows from Theorem B.5 by the Lasota-Yorke inequality, by the quasicompactness of ${\mathcal {L}}_\varphi$, and by the Perron-Frobenius-Ruelle theorem, which implies that ${\mathcal {L}}^N_\varphi h$ is comparable to $\int h d\mu$ for N large. The difference between the two can be controlled using $C^1$ bounds.

Step 2 We show that the oscillations in the sum induce enough cancellations.

Loosely, the argument goes as follows. We write, for a well chosen and large N:

$$\begin{aligned} \forall x \in U_b, \ {\mathcal {L}}^N_{\varphi - i (t \tau + l \theta )}h(x) = \underset{{\mathbf {a}} \rightsquigarrow b}{\sum _{{\mathbf {a}} \in {\mathcal {W}}_{N+1}} } e^{ i (t S_N \tau + l S_N \theta )(g_{\mathbf {a}}(x))} h(g_{\mathbf {a}}(x))e^{S_N \varphi (g_{\mathbf {a}}(x) ) } . \end{aligned}$$

If we choose x in a suitable open set ${\widehat{S}} \subset U_0$, the NLI and the NCP tell us that we can extract words ${\mathbf {a}}$ and ${\mathbf {b}}$ from this sum such that some cancellations happen. Indeed, if we isolate the term given by the words from the NLI,

$$\begin{aligned} e^{ i (t S_N \tau + l S_N \theta )(g_{\mathbf {a}}(x))} h(g_{\mathbf {a}}(x))e^{S_N \varphi (g_{\mathbf {a}}(x) ) } + e^{ i (t S_N \tau + l S_N \theta )(g_{\mathbf {b}}(x))} h(g_{\mathbf {b}}(x))e^{S_N \varphi (g_{\mathbf {b}}(x) ) } , \end{aligned}$$

we see that a difference in argument might give us some cancellations. The effect of h in the difference of argument can be carefully controlled by the $C^1$ norm of h. The interesting part comes from the complex exponential. The difference of arguments of this part is

$$\begin{aligned} t (S_N \tau \circ g_{\mathbf {a}} - S_N \tau \circ g_{\mathbf {b}}) + l (S_N \theta \circ g_{{\mathbf {a}}} - S_N \theta \circ g_{{\mathbf {b}}}) , \end{aligned}$$

which might be rewriten in the form

$$\begin{aligned} \langle (t,l) , ({\tilde{\tau }},{\tilde{\theta }}) \rangle . \end{aligned}$$

Then we proceed as follows. Choose a large number of points $(x_k)$ in $U_0$. If, for a given $x_k$, the difference of argument $\langle (t,l),({\tilde{\tau }},{\tilde{\theta }}) \rangle (x_k)$ is not large enough, we might use the NCP to construct another point $y_k$ next to $x_k$ such that $\langle (t,l),({\tilde{\tau }},{\tilde{\theta }}) \rangle (y_k)$ become larger. The construction goes as follows: the NLI ensures that $\nabla \langle (t,l) , ({\tilde{\tau }},{\tilde{\theta }}) \rangle (x_k) =: w_k \ne 0$. Hence, the direction ${\widehat{w}}_k := \frac{w_k}{|w_k|}$ is well defined. The NCP then ensures us the existence of some $y_k \in J$ which are very close to $x_k$ and such that $x_k - y_k$ is a vector pointing in a direction comparable to ${\widehat{w}}_k$. As we are following the gradient of $\langle (t,l) , ({\tilde{\tau }},{\tilde{\theta }}) \rangle$, we are sure that $\langle (t,l) , ({\tilde{\tau }},{\tilde{\theta }}) \rangle (y_k)$ will be larger than before.

We then let S be the set containing the points where the difference in argument is large enough, so it contains some $x_k$ and some $y_k$. This large enough difference in argument that is true in S is also true in a small open neighborhood ${\widehat{S}}$ of S.

We can then write, for $x \in {\widehat{S}}$, an inequality of the form:

$$\begin{aligned}&\left| e^{ i (t S_N \tau + l S_N \theta )(g_{\mathbf {a}}(x))} h(g_{\mathbf {a}}(x))e^{S_N \varphi (g_{\mathbf {a}}(x) ) } + e^{ i (t S_N \tau + l S_N \theta )(g_{\mathbf {b}}(x))} h(g_{\mathbf {b}}(x))e^{S_N \varphi (g_{\mathbf {b}}(x) ) } \right| \\&\quad \le (1-\eta ) |h(g_{\mathbf {a}}(x))| e^{S_N \varphi (g_{\mathbf {a}}(x))} + |h(g_{\mathbf {b}}(x))| e^{S_N \varphi (g_{\mathbf {b}}(x))} , \end{aligned}$$

where the $(1-\eta )$ comes in front of the part with the smaller modulus. This is, in spirit, lemma 5.2 of [OW17]. We can then summarize the information in the form of a function $\beta$ that is 1 most of the time, but that is less than $(1-\eta )^{1/2}$ on ${\widehat{S}}$. This allows us to write the following bound:

$$\begin{aligned} {\mathcal {L}}_{\varphi -i(t \tau + l \theta )}^N h \le {\mathcal {L}}_{\varphi }^N \left( |h| \beta \right) . \end{aligned}$$

One of the main difficulty of this part is to make sure that ${\widehat{S}}$ is a set of large enough measure, while still managing not to make the $C^1$-norm of $\beta$ explode. All the hidden technicalities in this part forces us to only get this bound for a well chosen N.

Step 3 These cancellations allow us to compare ${\mathcal {L}}$ to an operator that is contracting on a cone.

We define the following cone, on which the soon-to-be-defined Dolgopyat operator will be well behaved. Define

$$\begin{aligned} K_R({\mathfrak {U}}) := \{ H \in C^1_b({\mathfrak {U}}) \ | \ H \text { is positive, and} \ |\nabla H| \le R H \} , \end{aligned}$$

and then define the Dolgopyat operator by ${\mathcal {M}} H := {\mathcal {L}}_\varphi ^N (H \beta )$. We then show that, if $H \in K_R({\mathfrak {U}})$ (for a well chosen R):

• 1. ${\mathcal {M}}(H) \in K_R({\mathfrak {U}})$

• 2. $\Vert {\mathcal {M}}(H) \Vert _{L^2(\mu )}^2 \le (1-\varepsilon ) \Vert H \Vert _{L^2(\mu )}^2$.

The first point is done using the Lasota-Yorke inequalities, see lemma 5.1 in [OW17]. The second point goes, loosely, as follows.

We write, using Cauchy-Schwartz:

$$\begin{aligned} ({\mathcal {M}}H)^2 = {\mathcal {L}}_\varphi ^N(H \beta )^2 \le {\mathcal {L}}^N_\varphi (H^2){\mathcal {L}}^N_\varphi (\beta ^2) . \end{aligned}$$

On ${\widehat{S}}$, the cancellations represented in the function $\beta$ spread, thanks to the fact that $\varphi$ is normalized, as follows:

$$\begin{aligned} {\mathcal {L}}_\varphi ^N (\beta ^2)= & {} \sum _{{\mathbf {a}}} e^{ S_N \varphi \circ g_{{\mathbf {a}}} } \beta ^2 \circ g_{\mathbf {a}} \\= & {} \sum _{{\mathbf {a}} \text { where } \beta =1 } e^{S_N \varphi \circ g_{\mathbf {a}} } \beta ^2 \circ g_{\mathbf {a}} + \sum _{{\mathbf {a}} \text { where } \beta \text { is smaller} } e^{S_N \varphi \circ g_{\mathbf {a}} } \beta ^2 \circ g_{\mathbf {a}} \\\le & {} \sum _{{\mathbf {a}} \text { where } \beta =1 } e^{S_N \varphi \circ g_{\mathbf {a}} } + \sum _{{\mathbf {a}} \text { where } \beta \text { is smaller} } e^{S_N \varphi \circ g_{\mathbf {a}} } (1-\eta ) \\= & {} 1 - \eta e^{-N \Vert \varphi \Vert _\infty }. \end{aligned}$$

Then, we use the doubling property of $\mu =\sum _a \mu _a$ and the control given by the fact that $H \in K_R({\mathfrak {U}})$ to bound the integral on all J by the integral on ${\widehat{S}}$:

$$\begin{aligned} \int _J {\mathcal {L}}^N_\varphi (H^2) d\mu \le C_0 \int _{{\widehat{S}}} {\mathcal {L}}^N_\varphi (H^2) d\mu . \end{aligned}$$

Hence, we can write, using the fact that ${\mathcal {L}}_\varphi$ preserves $\mu$ and the previously mentioned Cauchy-Schwartz inequality:

$$\begin{aligned} \Vert H\Vert _{L^2(\mu )}^2 - \Vert {\mathcal {M}}H\Vert _{L^2(\mu )}^2\ge & {} \int _J \left( {\mathcal {L}}_\varphi ^N(H^2) - {\mathcal {L}}_\varphi ^N(H^2) {\mathcal {L}}_\varphi ^N( \beta ^2) \right) d\mu \\\ge & {} \int _{{\widehat{S}}} \left( {\mathcal {L}}_\varphi ^N(H^2) - {\mathcal {L}}_\varphi ^N(H^2) {\mathcal {L}}_\varphi ^N( \beta ^2) \right) d\mu \\\ge & {} \eta e^{-N \Vert \varphi \Vert _\infty } \int _{{\widehat{S}}}{\mathcal {L}}_\varphi ^N(H^2) d\mu \\\ge & {} \eta C_0^{-1} e^{-N \Vert \varphi \Vert _\infty } \Vert H \Vert _{L^2(\mu )}^2 := \varepsilon \Vert H \Vert _{L^2(\mu )}^2 . \end{aligned}$$

Hence

$$\begin{aligned} \Vert {\mathcal {M}}H\Vert _{L^2(\mu )}^2 \le (1-\varepsilon ) \Vert H\Vert _{L^2(\mu )}^2 . \end{aligned}$$

Step 4 We conclude by an iterative argument.

To conclude, we need to see that we may bound h by some $H \in K_R({\mathfrak {U}})$, and also that the contraction property is true for all n, not just N.

For any $n=kN$, we can inductively prove our bound. If $k=1$, we can choose $H_0:= \Vert h\Vert _{|t|+|l|}$. Then, $|h| \le H_0$ and so

$$\begin{aligned} \Vert {\mathcal {L}}_{\varphi - i(t \tau + l \theta ) }^N h \Vert _{L^2(\mu )} \le \Vert {\mathcal {M}} H_0 \Vert _{L^2(\mu )} \le (1-\varepsilon )^{1/2} \Vert h\Vert _{(|t|+|l|)} . \end{aligned}$$

Then, choosing $H_{k+1} := {\mathcal {M}} H_k \in K_R({\mathfrak {U}})$, we can proceed to the next step of the induction and get

$$\begin{aligned} \Vert {\mathcal {L}}_{\varphi - i(t \tau + l \theta ) }^{kN} h \Vert _{L^2(\mu )} \le \Vert {\mathcal {M}} H_{k-1} \Vert _{L^2(\mu )} \le (1-\varepsilon )^{k/2} \Vert h\Vert _{(|t|+|l|)} . \end{aligned}$$

Finally, if $n=kN+r$, with $0 \le j \le N-1$, we write

$$\begin{aligned} \Vert {\mathcal {L}}_{\varphi - i(t \tau + l \theta ) }^{kN+r} h \Vert _{L^2(\mu )} \le (1-\varepsilon )^k \Vert {\mathcal {L}}_\varphi ^r h\Vert _{(|t|+|l|)} \lesssim (1-\varepsilon )^{n/(2N)} \Vert h\Vert _{(|t|+|l|)} , \end{aligned}$$

and the proof is done.