Statistical properties of mostly contracting fast-slow partially hyperbolic systems

Abstract

We consider a class of ${\mathcal C}^{4}$ partially hyperbolic systems on ${\mathbb T}^2$ described by maps $F_\varepsilon (x,\theta )=(f(x,\theta ),\theta +\varepsilon \omega (x,\theta ))$ where $f(\cdot ,\theta )$ are expanding maps of the circle. For sufficiently small $\varepsilon$ and $\omega$ generic in an open set, we precisely classify the SRB measures for $F_\varepsilon$ and their statistical properties, including exponential decay of correlation for Hölder observables with explicit and nearly optimal bounds on the decay rate.

Appendix: Random walks

We start by recalling a well known fact about one dimensional random walks (it can be obtained, e.g., from Cramer’s Theorem).

Lemma 11.1

Let $\xi _k\in \{-1,1\}$ be a sequence of i.i.d. random variables with distribution ${\mathbb P}\left( \xi _i=1\right) =p$ for $p\in (0,1)$. Let $\mathbf {\Xi }_0=0$ and for $n>0,$ define :  $\mathbf {\Xi }_n=\sum _{j=1}^n\xi _j$. For any $c<2p-1$ there exist $\vartheta ,\varrho \in (0,1)$ such that,  for any $k\in {\mathbb N}$ and $a\in {\mathbb R}{:}$

$$\begin{aligned} {\mathbb P}\left( \mathbf {\Xi }_k \le kc - a\right) \le \varrho ^a\vartheta ^k. \end{aligned}$$

Next, we introduce an useful comparison argument:

Lemma 11.2

Let $\xi _k\in \{-1,1\}$ be a sequence of independent random variables and let $\eta _k\in \{-1,0,1\}$ be a random process such that

$$\begin{aligned} {\mathbb P}\left( \eta _{k+1}=1|\eta _1\cdots \eta _{k}\right) \ge {\mathbb P}\left( \xi _{k+1}=1\right) . \end{aligned}$$

For $n>0$ define the random variables

$$\begin{aligned} \mathbf {\Xi }_n=\sum _{j=1}^n\xi _j\quad \mathbf {H}_n=\sum _{j=1}^n\eta _j \end{aligned}$$

where $N>0$ is some fixed natural number (if $n=0$ we let them all equal to 0);  then for each $n\in {\mathbb N}$ and $L\in {\mathbb Z}{:}$

$$\begin{aligned} {\mathbb P}\left( \mathbf {H}_{k}\le L\right) \le {\mathbb P}\left( \mathbf {\Xi }_{k}\le L\right) . \end{aligned}$$

(11.1)

In particular,  if $\tau _\mathbf {\Xi }$ is the hitting time $\tau =\inf \{k\,:\,\mathbf {\Xi }_k \ge L\}$ and $\tau _\mathbf {H}=\inf \{k\,:\,\mathbf {H}_k \ge L\}$ we have,  for any $s>0{:}$

$$\begin{aligned} {\mathbb P}\left( \tau _\mathbf {H}> s\right) \le {\mathbb P}\left( \tau _\mathbf {\Xi }> s\right) \end{aligned}$$

Proof

(see [17, Proposition 2.4]) The proof amounts to design a suitable coupling $(\xi ^*_k,\eta ^*_k)$ of the random variables $\xi _k$ and $\eta _k$. Let us introduce an auxiliary sequence $U_k$ of independent random variables uniformly distributed on [0, 1] and define the random variables

$$\begin{aligned} \xi ^*_k= {\left\{ \begin{array}{ll} +1 &{}\mathrm{if}\ U_k<{\mathbb P}\left( \xi _k\ge 1\right) \\ -1&{}\mathrm{otherwise} \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} \eta ^*_k= {\left\{ \begin{array}{ll} +1 &{}\mathrm{if}\ U_k<{\mathbb P}\left( \eta _k= 1|\eta _1=\eta ^*_1,\ldots ,\eta _{k-1}=\eta ^*_{k-1}\right) \\ -1 &{}\mathrm{if}\ U_k\ge 1-{\mathbb P}\left( \eta _k = -1|\eta _1=\eta ^*_1,\ldots ,\eta _{k-1}=\eta ^*_{k-1}\right) \\ 0&{}\mathrm{otherwise}. \end{array}\right. } \end{aligned}$$

We then define

$$\begin{aligned} \mathbf {\Xi }^*_n=\sum _{j=1}^n\xi ^*_{j};\quad \mathbf {H}^*_n=\sum _{j=1}^n\eta ^*_j \end{aligned}$$

Clearly $\xi ^*_k$ (resp. $\eta ^*_k$) has the same distribution of $\xi _k$ (resp. $\eta _k$) and consequently $\mathbf {\Xi }^*_k$ (resp. $\mathbf {H}^*_k$) has the same distribution of $\mathbf {\Xi }_k$ (resp. $\mathbf {H}_k$). Moreover, $\xi ^*_k\le \eta ^*_k$ by design which in turn implies that $\mathbf {\Xi }^*_k\le \mathbf {H}^*_k$. This concludes the proof of our lemma.$\square$