∎

11institutetext: Leonid A. Bunimovich 22institutetext: School of Mathematics,
Georgia Institute of Technology, Atlanta, USA
22email<EMAIL_ADDRESS>33institutetext: Yaofeng Su
44institutetext: School of Mathematics, Georgia Institute of Technology,
Atlanta, USA
44email<EMAIL_ADDRESS>
# Estimates of constants in the limit theorems for chaotic dynamical systems

Leonid A. Bunimovich  Yaofeng Su

(Received: date / Accepted: date)

###### Abstract

In a vast area of probabilistic limit theorems for dynamical systems with
chaotic behaviors always only functional form (exponential, power, etc) of the
asymptotic laws and of convergence rates were studied. However, for basically
all applications, e.g., for computer simulations, development of algorithms to
study chaotic dynamical systems numerically, as well as for design and
analysis of real (e.g., in physics) experiments, the exact values (or at least
estimates) of constants (parameters) of the functions, which appear in the
asymptotic laws and rates of convergence, are of primary interest. In this
paper we provide such estimates of constants (parameters) in the central limit
theorem, large deviations principle, law of large numbers and the rate of
correlations decay for strongly chaotic dynamical systems.

###### Contents

  1. 1 Introduction
  2. 2 Definitions and Notations
  3. 3 Proofs
  4. 4 Applications

##  1 Introduction

Dynamical systems with an irregular (“complex”) behavior (loosely called
chaotic systems) were discovered in the last century. Their evolution (in
“time”) looks similar to the evolution of random processes. Since the
remarkable Kolmogorov’s paper Kolmogorov [1958] which built a “bridge” between
the worlds of deterministic (dynamical) systems and random systems (stochastic
processes), a natural question was raised whether some classes of chaotic
dynamical systems satisfy limit theorems known for stochastic processes. One
should mention that ergodicity (which is equivalent to the strong law of large
numbers) was established for various dynamical systems. However, non-chaotic
systems can be ergodic too. Perhaps, the most famous (and somewhat simple)
example of ergodic, but non-chaotic, system is the irrational rotation of a
circle. Therefore one should generally impose on a dynamical system some
stronger, than ergodicity, statistical properties. Starting with the second
half of the last century studies of limit theorems for chaotic dynamical
systems became one of the most active areas in the theory of dynamical
systems. The results obtained in this area allowed for many breakthroughs in
various applications of dynamical systems. One of the most explored areas was
related to the problem of the existence of various transport coefficients in
nonequilibrium statistical mechanics, which requires estimations of the rates
of correlations decay for some meaningful observables. The results in this
area deal with establishing a functional form of the limits of various
averages along the orbits of dynamical systems, or of the rates of convergence
to these limits. For instance, for existence of transport coefficients it is
enough to prove that correlations of some specific functions (observables) on
the phase space of a system in question are integrable over an infinite time
interval, i.e., the correlations decay sufficiently fast. All estimates in
these limit theorems contain various constants, e.g., the exponential estimate
looks like $a\cdot\exp(b\cdot t)$. However, what are the values of $a$ and $b$
in this expression? Such questions are of importance for applications of the
dynamical systems theory in any areas of science and technology, particularly
for numerical studies and simulations of chaotic dynamical systems, for
developing the corresponding algorithms, and even for building relevant
devices for experimental studies of chaotic systems. In the present paper we
obtain estimates of constants appearing in main limit theorems of probability
theory like the central limit theorem, the large deviation principle, the law
of large numbers and the rate of correlations decay.

The organization of the paper: in section 2 we first introduce the
definitions, notations, and main theorems. In section 3, we prove these
theorems. We conclude this paper with applications in section 4.

##  2 Definitions and Notations

######  Definition 1 (Topological Markov shifts).

The phase space $\mathbb{A}$ of a topological Markov shift with finite states
space $S$ is defined as

$\mathbb{A}:=\\{(x_{0},x_{1},x_{2}\cdots):x_{i}\in S,\quad\\#S<\infty\\}.$

The dynamics on $\mathbb{A}$ is the left shift
$\sigma:\mathbb{A}\to\mathbb{A}:\sigma(x_{0},x_{1},\cdots)=(x_{1},x_{2},\cdots)$.
Let there is $\theta\in(0,1)$, such that for any two points
$x,y\in\mathbb{A}$, the distance $d(x,y)$ between them equals

$d(x,y):=\theta^{\inf\\{n\geq 0:\text{ }x_{n}\neq y_{n}\\}}.$

Consider a potential $\phi_{p}:\mathbb{A}\to\mathbb{R}$, which is a Lipshitz
function. The last means that there is $||\phi_{p}||\geq 1$, such that

$||\phi_{p}||_{\infty}\leq||\phi_{p}||,\quad|\phi_{p}(x)-\phi_{p}(y)|\leq||\phi_{p}||d(x,y).$

We also suppose that $\phi_{p}$ is normalized, i.e., it satisfies the relation

$\sum_{a\in S}e^{\phi_{p}(ax)}=1\text{ for any }x\in\mathbb{A}.$

An equilibrium state is a probability measure $\mu_{\phi_{p}}$, which is
shift-invariant (i.e. $\mu_{\phi_{p}}\circ\sigma^{-1}=\mu_{\phi_{p}}$). Denote
the expectation with respect to $\mu_{\phi_{p}}$ by $\mathbb{E}$.

Next, a transfer operator is defined as

$P(\phi)(x)=\sum_{a\in S}e^{\phi_{p}(ax)}\phi(ax)$

In what follows, we will always denote an observable for the dynamical system
$(\mathbb{A},\sigma,\mu_{\phi_{p}})$ by $\phi:\mathbb{A}\to\mathbb{R}$. We
also assume that all observables under consideration are Lipschitz, i.e.,
there exists $||\phi||\geq 1$, such that

$||\phi||_{\infty}\leq||\phi||,\quad|\phi(x)-\phi(y)|\leq||\phi||d(x,y).$

######  Remark 1.

For uniformly hyperbolic (e.g., the Axiom A) systems, $\theta$ is the rate of
expansion along unstable manifolds, and $||\phi_{p}||$ is the constant
appearing in the expression of the distortion (ratio of the expansion rates)
along unstable manifolds.

######  Remark 2.

To estimate the constants appearing in the exponential mixing rates we will
use the maximal coupling techniques Bressaud et al. [1999].

The main results of the present paper are listed below.

######  Theorem 2.1 (Estimates of an error in the rate of correlations decay).

Consider the topological dynamical system in Definition 1. We have

$||P^{n}(\phi-\mathbb{E}\phi)||_{\infty}\leq\frac{(\sqrt{z_{0}})^{-n}}{(\log\sqrt{z_{0}})(1-a-\epsilon)}||\phi||,$

where $\epsilon\in(0,\min\\{1-a,1-\theta\\})$,
$a=1-\exp(-||\phi_{p}||(1-\theta)^{-1})$ and

$z_{0}\in\Big{(}1,\min\Big{\\{}\Big{(}\frac{a+\epsilon/2}{a}\Big{)}^{U},\frac{\theta+\epsilon}{\theta},\frac{1-\epsilon}{\theta},e^{2}\Big{\\}}\Big{)},\quad
U:=\frac{\log(1-\epsilon)}{\log[\theta\epsilon(1-\epsilon-\theta)]-\log(2||\phi_{p}||)}$

######  Theorem 2.2 (Estimates of an error in the CLT).

Consider the topological dynamical system in Definition 1. Suppose that the
observable $\phi$ has zero mean, and let
$\Sigma^{2}:=\mathbb{E}\phi^{2}+2\sum_{i\geq 1}\mathbb{E}\phi\phi^{i}$. Then
for any $t\in\mathbb{R}$

$\Big{|}\mathbb{E}\exp\Big{(}it\frac{\sum_{0\leq i\leq
n-1}\phi\circ\sigma^{i}}{\sqrt{n}}\Big{)}-\exp\Big{(}-\frac{t^{2}\Sigma^{2}}{2}\Big{)}\Big{|}\leq
C_{t,n},$

where

$\displaystyle C_{t,n}=$ $\displaystyle
6922\Big{(}\frac{|t|^{4}||\phi||^{4}}{n^{0.1}(1-z_{0}^{-0.5})^{4}(1-a-\epsilon)^{4}(\ln\sqrt{z_{0}})^{4}}+\frac{t^{2}z_{0}^{-0.5n}||\phi||^{2}}{(1-z_{0}^{-0.5})(1-a-\epsilon)\ln\sqrt{z_{0}}}\Big{)}$
$\displaystyle+64\Big{(}\frac{t^{2}n^{3}||\phi||^{2}e^{2||\phi_{p}||\theta(1-\theta)^{-1}}||\phi_{p}||^{2}(\sqrt{z_{0}})^{-n^{0.2}}}{(1-\theta)^{2}(1-a-\epsilon)\ln\sqrt{z_{0}}}\Big{)},$

and $a,\epsilon,z_{0}$ are the ones in Theorem 2.1.

######  Theorem 2.3 (Estimates of an error in the large deviations principle).

Consider the topological dynamical system in Definition 1. Suppose again that
$\phi$ has zero mean. Then for any $u>0$

$\displaystyle\mu_{\phi_{p}}\Big{(}\Big{|}\frac{\sum_{i\leq
n-1}\phi\circ\sigma^{i}}{n}\Big{|}\geq u\Big{)}$ $\displaystyle\leq
2\exp\Big{(}\frac{u(1-z_{0}^{-0.5})(1-a-\epsilon)(\ln\sqrt{z_{0}})}{36||\phi||z_{0}^{0.5}}\Big{)}\exp\Big{(}-\frac{u^{2}n(1-z_{0}^{-0.5})^{2}(1-a-\epsilon)^{2}(\ln\sqrt{z_{0}})^{2}}{72||\phi||^{2}}\Big{)},$

where $a,\epsilon,z_{0}$ are the ones in Theorem 2.1.

######  Remark 3.

Note that a stronger form of the large deviation principle is proved in
Bunimovich and Su [2022].

######  Theorem 2.4 (Estimates of an error in the law of large numbers).

Consider the topological dynamical system in Definition 1. Suppose that $\phi$
has zero mean. Then there exists a sufficiently small $\delta\in(0,0.5)$, such
that for a.e. $x\in\mathbb{A}$, any $n\geq N_{x,\delta},$

$\Big{|}\frac{\sum_{i\leq n}\phi\circ\sigma^{i}(x)}{n}\Big{|}\leq
n^{-(0.5-\delta)},$

where $N_{x,\delta}:=\sum_{N\geq 0}\mathbbm{1}_{\bigcup_{n\geq N}|\sum_{i\leq
n-1}\phi\circ\sigma^{i}(x)|\geq n^{0.5+\delta}}$ satisfies the relation
$||N_{x,\delta}||_{1}$

$\leq
2\exp\Big{(}\frac{(1-z_{0}^{-0.5})(1-a-\epsilon)(\ln\sqrt{z_{0}})}{36||\phi||z_{0}^{0.5}}\Big{)}\sum_{m\geq
1}m\exp\Big{(}-\frac{m^{2\delta}(1-z_{0}^{-0.5})^{2}(1-a-\epsilon)^{2}(\ln\sqrt{z_{0}})^{2}}{72||\phi||^{2}}\Big{)},$

and $a,\epsilon,z_{0}$ are the ones in Theorem 2.1.

##  3 Proofs

###### A proof of Theorem 2.1.

Consider a topological Markov shift in Definition 1. Recall now the Theorem 1
of Bressaud et al. [1999], which says that

$|\int f\cdot
P^{n}(\phi-\mathbb{E}\phi)d\mu_{\phi_{p}}|\leq||f||_{1}\frac{||\phi||}{||\phi_{p}||}\sum_{k=0}^{n}||\phi_{p}||\theta^{k}\gamma_{n-k}^{*},$

therefore,

$||P^{n}(\phi-\mathbb{E}\phi)||_{\infty}\leq||\phi||\sum_{k=0}^{n}\theta^{k}\gamma_{n-k}^{*},$

where $\gamma_{k}^{*}:=\mathbb{P}(S_{k}=0)$, $(S_{n})_{n\geq 0}$ is a Markov
chain with values in $\mathbb{N}\bigcup\\{0\\}$, which starts from $0$, and
has transition probabilities $p_{k,k+1}=1-\gamma_{k}$, $p_{k,0}=\gamma_{k}$,
and $\gamma_{k}:=1-\exp(-||\phi_{p}||\theta^{k})$.

We compute now $\gamma_{k}^{*}$. It follows from (A.3)-(A.5) of Bressaud et
al. [1999] that

$\sum_{k\geq 1}\mathbb{P}(\tau=k)<1,$

and

$\sum_{k}\mathbb{P}(S_{k}=0)z_{0}^{k}=\frac{1}{1-\sum_{k\geq
1}z_{0}^{k}\mathbb{P}(\tau=k)}\text{ for some }z_{0}>1,$

where

$\mathbb{P}(\tau=k)=\mathbb{P}(\inf\\{n>0:S_{n}=0\\}=k)=\prod_{i=0}^{k-2}p_{i,i+1}p_{k-1,0}\leq||\phi_{p}||\theta^{k-1},$
$\mathbb{P}(\tau=k)=\prod_{i=0}^{k-2}p_{i,i+1}p_{k-1,0}=\prod_{i=0}^{k-2}p_{i,i+1}-\prod_{i=0}^{k-1}p_{i,i+1}.$

Choose now $a,z_{0},U$ and $\epsilon$ as in the formulation of the Theorem
2.1, let $N=U^{-1}$. Then

$\displaystyle\sum_{i\geq 1}\mathbb{P}(\tau=i)z_{0}^{i}$
$\displaystyle=\sum_{i\geq
N}\mathbb{P}(\tau=i)z_{0}^{i}+\sum_{i<N}\mathbb{P}(\tau=i)z_{0}^{i}$
$\displaystyle\leq\sum_{i\geq
N}\frac{||\phi_{p}||}{\theta}(z_{0}\theta)^{i}+\sum_{i<N}\mathbb{P}(\tau=i)z_{0}^{i}$
$\displaystyle\leq\sum_{i\geq
N}\frac{||\phi_{p}||}{\theta}(z_{0}\theta)^{i}+\sum_{i\geq
1}\mathbb{P}(\tau=i)z_{0}^{N}$
$\displaystyle\leq\frac{||\phi_{p}||}{\theta}\frac{(z_{0}\theta)^{N}}{1-z_{0}\theta}+\Big{[}1-\exp\Big{(}-\frac{||\phi_{p}||}{1-\theta}\Big{)}\Big{]}z_{0}^{N}\leq
a+\epsilon.$

Therefore $\mathbb{P}(S_{k}=0)z_{0}^{k}\leq\frac{1}{1-a-\epsilon}$, and

$||P^{n}(\phi-\mathbb{E}\phi)||_{\infty}\leq||\phi||\sum_{k=0}^{n}\theta^{k}\gamma_{n-k}^{*}\leq||\phi||\frac{nz_{0}^{-n}}{1-a-\epsilon}\leq||\phi||\frac{(\sqrt{z_{0}})^{-n}}{(1-a-\epsilon)\log\sqrt{z_{0}}},$

which implies the statement of Theorem 2.1. ∎

###### Proof of Theorem 2.2.

We adopt the martingale methods used in Su . (See also De Simoi and Liverani
[2015], where martingale methods were applied to analysis of partially
hyperbolic dynamical systems). Let $\mathcal{B}$ be the $\sigma$-algebra of
$\mathbb{A}$, $\mathcal{F}_{n}:=\sigma^{-n}\mathcal{B}$ and

$H_{n}\circ\sigma^{n}:=\mathbb{E}\Big{[}\sum_{i\leq
n-1}\phi\circ\sigma^{i}|\mathcal{F}_{n}\Big{]},$
$\psi_{n}\circ\sigma^{n}:=\phi\circ\sigma^{n}+H_{n}\circ\sigma^{n}-H_{n+1}\circ\sigma^{n+1}$

Then $H_{n}=\sum_{i\leq n-1}P^{n-i}\phi$, $(\psi_{n}\circ\sigma^{n})_{n\geq
0}$ are reverse martingale differences and

$\sum_{n\leq N}\psi_{n}\circ\sigma^{n}=\sum_{n\leq
N}\phi\circ\sigma^{n}-H_{N+1}\circ\sigma^{N+1}.$ (3.1)

By applying Theorem 2.1, we get, for any $n\geq 1$, that

$||P^{n}\phi||_{\infty}\leq||\phi||,\quad\sup_{n}||H_{n}||_{\infty}\leq||\phi||\frac{z_{0}^{-0.5}}{(1-z_{0}^{-0.5})(1-a-\epsilon)\ln\sqrt{z_{0}}},$
(3.2)
$||\psi_{n}||_{\infty}\leq||\phi||_{\infty}+||H_{n}||_{\infty}+||H_{n+1}\circ\sigma||_{\infty}\leq\frac{3||\phi||}{(1-z_{0}^{-0.5})(1-a-\epsilon)\ln\sqrt{z_{0}}},$
(3.3)

$\Sigma^{2}=\lim_{N\to\infty}\frac{\mathbb{E}(\sum_{n\leq
N}\phi\circ\sigma^{n})^{2}}{N}=\lim_{N\to\infty}\frac{\sum_{n\leq
N}\mathbb{E}\psi_{n}^{2}\circ\sigma^{n}}{N},$
$\Big{|}\frac{\mathbb{E}(\sum_{n\leq
N}\phi\circ\sigma^{n})^{2}}{N}-\frac{\sum_{n\leq
N}\mathbb{E}\psi_{n}^{2}\circ\sigma^{n}}{N}\Big{|}=\frac{\mathbb{E}H^{2}_{N+1}}{N}\leq\frac{||\phi||^{2}z_{0}^{-1}}{N(1-z_{0}^{-0.5})^{2}(1-a-\epsilon)^{2}(\ln\sqrt{z_{0}})^{2}}.$

By making use of Theorem 2.1, and of the argument in the proof of Corollary
3.10 in Su , we obtain the following relations

$\Big{|}\frac{\mathbb{E}(\sum_{n\leq
N}\phi\circ\sigma^{n})^{2}}{N}-\Sigma^{2}\Big{|}\leq||\phi||^{2}\frac{z_{0}^{-0.5}}{N(1-z_{0}^{-0.5})^{2}(1-a-\epsilon)\ln\sqrt{z_{0}}}+\frac{z_{0}^{-0.5(N+1)}||\phi||^{2}}{(1-z_{0}^{-0.5})(1-a-\epsilon)\ln\sqrt{z_{0}}},$
$\Big{|}\frac{\sum_{n\leq
N}\mathbb{E}\psi_{n}^{2}\circ\sigma^{n}}{N}-\Sigma^{2}\Big{|}\leq\frac{2||\phi||^{2}z_{0}^{-0.5}}{N(1-z_{0}^{-0.5})^{2}(1-a-\epsilon)^{2}(\ln\sqrt{z_{0}})^{2}}+\frac{z_{0}^{-0.5(N+1)}||\phi||^{2}}{(1-z_{0}^{-0.5})(1-a-\epsilon)\ln\sqrt{z_{0}}}.$
(3.4)

Next we address a regularity of $\psi_{n}$. One has

$\displaystyle|(P^{n}\phi)(x_{1})-(P^{n}\phi)(x_{2})|=\big{|}\sum_{\sigma^{n}y_{1}=x_{1}}e^{\sum_{i\leq
n-1}\phi_{p}(\sigma^{i}y_{1})}\phi(y_{1})-\sum_{\sigma^{n}y_{2}=x_{2}}e^{\sum_{i\leq
n-1}\phi_{p}(\sigma^{i}y_{2})}\phi(y_{2})\big{|}$ $\displaystyle=\big{|}\sum
e^{\sum_{i\leq n-1}\phi_{p}(\sigma^{i}y_{2})}\phi(y_{1})[e^{\sum_{i\leq
n-1}\phi_{p}(\sigma^{i}y_{1})-\phi_{p}(\sigma^{i}y_{2})}-1]+\sum
e^{\sum_{i\leq n-1}\phi_{p}(\sigma^{i}y_{2})}[\phi(y_{1})-\phi(y_{2})]\big{|}$
$\displaystyle\leq|\sum e^{\sum_{i\leq
n-1}\phi_{p}(\sigma^{i}y_{2})}\phi(y_{1})[e^{\sum_{i\leq
n-1}\phi_{p}(\sigma^{i}y_{1})-\phi_{p}(\sigma^{i}y_{2})}-1]|+||\phi||\theta^{n}d(x_{1},x_{2})$
$\displaystyle\leq||\phi||[e^{||\phi_{p}||d(x_{1},x_{2})\theta(1-\theta)^{-1}}-1]+||\phi||\theta^{n}d(x_{1},x_{2})$
$\displaystyle\leq||\phi||[e^{||\phi_{p}||\theta(1-\theta)^{-1}}||\phi_{p}||\theta(1-\theta)^{-1}+\theta^{n}]d(x_{1},x_{2})\leq
2\theta||\phi||e^{||\phi_{p}||\theta(1-\theta)^{-1}}||\phi_{p}||(1-\theta)^{-1}d(x_{1},x_{2}).$

Therefore, $||P^{n}\phi||\leq
2||\phi||e^{||\phi_{p}||\theta(1-\theta)^{-1}}||\phi_{p}||(1-\theta)^{-1}$,
and

$||H_{n}||\leq
2n||\phi||e^{||\phi_{p}||\theta(1-\theta)^{-1}}||\phi_{p}||(1-\theta)^{-1},$
$||\psi_{n}||\leq||\phi||+||H_{n}||+||H_{n+1}\circ\sigma||\leq(4n+3)||\phi||e^{||\phi_{p}||\theta(1-\theta)^{-1}}||\phi_{p}||(1-\theta)^{-1}.$
(3.5)

######  Notation 3.1.

In order to simplify the proofs, we will write in what follows $f=O_{C}(g)$
instead of $|f|\leq C|g|$.

Next we estimate the following expression

$\Big{|}\mathbb{E}\exp\Big{(}it\frac{\sum_{0\leq i\leq
n-1}\phi\circ\sigma^{i}}{\sqrt{n}}\Big{)}-\exp\Big{(}-\frac{t^{2}\Sigma^{2}}{2}\Big{)}\Big{|}.$

Consider the blocks $\\{I_{i}:\mathbf{card}(I_{i})=n^{0.8},1\leq i\leq
n^{0.2}\\}$ of consecutive positive integers in $[0,n-1]$, and the blocks
$\\{I_{i,1},I_{i,2}:\mathbf{card}(I_{i,1})=n^{0.8}-n^{0.2},\mathbf{card}(I_{i,2})=n^{0.2}\\}$
of consecutive positive integers in each $I_{i}$, which have no gaps between
the blocks. Let $X_{i}:=\sum_{k\in I_{i,1}}\psi_{k}\circ\sigma^{k}$,
$\mathcal{F}_{i}=\sigma^{-in^{0.8}}\mathcal{B}$. Then, it follows from (3.1)
and (3.2), that

$\displaystyle\mathbb{E}\exp\Big{(}$ $\displaystyle it\frac{\sum_{0\leq i\leq
n-1}\phi\circ\sigma^{i}}{\sqrt{n}}\Big{)}=\mathbb{E}\exp\Big{(}it\frac{\sum_{0\leq
i\leq
n-1}\psi_{i}\circ\sigma^{i}}{\sqrt{n}}\Big{)}+O_{1}\Big{(}\frac{|t|||\phi||z_{0}^{-0.5}}{n^{0.5}(1-z_{0}^{-0.5})(1-a-\epsilon)\ln\sqrt{z_{0}}}\Big{)}$
$\displaystyle=\exp\Big{(}-\frac{t^{2}\sum_{i\leq
n-1}\mathbb{E}\psi_{i}^{2}\circ\sigma^{i}}{2n}\Big{)}+\mathbb{E}\exp\Big{(}it\frac{\sum_{0\leq
i\leq n-1}\psi_{i}\circ\sigma^{i}}{\sqrt{n}}\Big{)}$
$\displaystyle\quad-\exp\Big{(}-\frac{t^{2}\sum_{i\leq
n-1}\mathbb{E}\psi_{i}^{2}\circ\sigma^{i}}{2n}\Big{)}+O_{1}\Big{(}\frac{|t|||\phi||z_{0}^{-0.5}}{n^{0.5}(1-z_{0}^{-0.5})(1-a-\epsilon)\ln\sqrt{z_{0}}}\Big{)}.$

By making use of (3.3), we can now continue the estimates as

$\displaystyle=\exp\Big{(}-\frac{t^{2}\sum_{i\leq
n-1}\mathbb{E}\psi_{i}^{2}\circ\sigma^{i}}{2n}\Big{)}+\mathbb{E}\exp\Big{(}it\frac{\sum_{i=1}^{n^{0.2}}X_{i}}{\sqrt{n}}\Big{)}-\exp\Big{(}-\frac{t^{2}\sum_{i=1}^{n^{0.2}}\mathbb{E}X_{i}^{2}}{2n}\Big{)}$
$\displaystyle\quad+O_{3}\Big{(}\frac{|t|n^{0.4}||\phi||}{n^{0.5}(1-z_{0}^{-0.5})(1-a-\epsilon)\ln\sqrt{z_{0}}}\Big{)}+O_{5}\Big{(}\frac{|t|^{2}n^{0.6}||\phi||^{2}}{n(1-z_{0}^{-0.5})^{2}(1-a-\epsilon)^{2}(\ln\sqrt{z_{0}})^{2}}\Big{)}$
$\displaystyle\quad+O_{1}\Big{(}\frac{|t|||\phi||z_{0}^{-0.5}}{n^{0.5}(1-z_{0}^{-0.5})(1-a-\epsilon)\ln\sqrt{z_{0}}}\Big{)}$
$\displaystyle=\exp\Big{(}-\frac{t^{2}\sum_{i\leq
n-1}\mathbb{E}\psi_{i}^{2}\circ\sigma^{i}}{2n}\Big{)}+\mathbb{E}\exp\Big{(}it\frac{\sum_{i=1}^{n^{0.2}}X_{i}}{\sqrt{n}}\Big{)}-\exp\Big{(}-\frac{t^{2}\sum_{i=1}^{n^{0.2}}\mathbb{E}X_{i}^{2}}{2n}\Big{)}$
$\displaystyle\quad+O_{9}\Big{(}\frac{|t|^{2}||\phi||^{2}}{n^{0.1}(1-z_{0}^{-0.5})^{2}(1-a-\epsilon)^{2}(\ln\sqrt{z_{0}})^{2}}\Big{)}.$

We will use the Taylor expansion, as in the proof of Corollary 3.1 of Su , and
apply the Burkholder–Davis–Gundy inequalities to $X_{i}$, we can further
continue the estimates as

$\displaystyle=\exp\Big{(}-\frac{t^{2}\sum_{i\leq
n-1}\mathbb{E}\psi_{i}^{2}\circ\sigma^{i}}{2n}\Big{)}+\sum_{i=1}^{n^{0.2}}O_{1}\Big{(}\mathbb{E}\Big{|}\frac{tX_{i}}{n^{0.5}}\Big{|}^{3}+\Big{|}\frac{t^{2}\mathbb{E}X^{2}_{i}}{n}\Big{|}^{2}\Big{)}$
$\displaystyle\quad+\sum_{i=1}^{n^{0.2}}O_{1}\Big{(}\frac{t^{2}\mathbb{E}|\mathbb{E}[X^{2}_{i}-\mathbb{E}X^{2}_{i}|\mathcal{F}_{i}]|}{2n}\Big{)}+O_{9}\Big{(}\frac{|t|^{2}||\phi||^{2}}{n^{0.1}(1-z_{0}^{-0.5})^{2}(1-a-\epsilon)^{2}(\ln\sqrt{z_{0}})^{2}}\Big{)}$
$\displaystyle=\exp\Big{(}-\frac{t^{2}\sum_{i\leq
n-1}\mathbb{E}\psi_{i}^{2}\circ\sigma^{i}}{2n}\Big{)}+\sum_{i=1}^{n^{0.2}}O_{64}\Big{(}\mathbb{E}(\sum_{k\in
I_{i,1}}\psi_{k}^{2}\circ\sigma^{k})^{1.5}\Big{|}\frac{t}{n^{0.5}}\Big{|}^{3}+\Big{|}\frac{t^{2}\mathbb{E}X^{2}_{i}}{n}\Big{|}^{2}\Big{)}$
$\displaystyle\quad+\sum_{i=1}^{n^{0.2}}O_{1}\Big{(}\frac{t^{2}\mathbb{E}|\mathbb{E}[X^{2}_{i}-\mathbb{E}X^{2}_{i}|\mathcal{F}_{i}]|}{2n}\Big{)}+O_{9}\Big{(}\frac{|t|^{2}||\phi||^{2}}{n^{0.1}(1-z_{0}^{-0.5})^{2}(1-a-\epsilon)^{2}(\ln\sqrt{z_{0}})^{2}}\Big{)}.$

Now, from (3.3) and (3.4), we get that the previous expression equals

$\displaystyle=\exp\Big{(}-\frac{t^{2}\sum_{i\leq
n-1}\mathbb{E}\psi_{i}^{2}\circ\sigma^{i}}{2n}\Big{)}+\sum_{i=1}^{n^{0.2}}O_{6912}\Big{(}\frac{|t|^{4}||\phi||^{4}}{n^{0.3}(1-z_{0}^{-0.5})^{4}(1-a-\epsilon)^{4}(\ln\sqrt{z_{0}})^{4}}\Big{)}$
$\displaystyle\quad+\sum_{i=1}^{n^{0.2}}O_{1}\Big{(}\frac{t^{2}\mathbb{E}|\mathbb{E}[X^{2}_{i}-\mathbb{E}X^{2}_{i}|\mathcal{F}_{i}]|}{2n}\Big{)}+O_{9}\Big{(}\frac{|t|^{2}||\phi||^{2}}{n^{0.1}(1-z_{0}^{-0.5})^{2}(1-a-\epsilon)^{2}(\ln\sqrt{z_{0}})^{2}}\Big{)}$
$\displaystyle=\exp\Big{(}-\frac{t^{2}\sum_{i\leq
n-1}\mathbb{E}\psi_{i}^{2}\circ\sigma^{i}}{2n}\Big{)}+\sum_{i=1}^{n^{0.2}}O_{1}\Big{(}\frac{t^{2}\mathbb{E}|\mathbb{E}[X^{2}_{i}-\mathbb{E}X^{2}_{i}|\mathcal{F}_{i}]|}{n}\Big{)}$
$\displaystyle\quad+O_{6921}\Big{(}\frac{|t|^{4}||\phi||^{4}}{n^{0.1}(1-z_{0}^{-0.5})^{4}(1-a-\epsilon)^{4}(\ln\sqrt{z_{0}})^{4}}\Big{)}$
$\displaystyle=\exp\Big{(}-\frac{t^{2}\Sigma^{2}}{2}\Big{)}+O_{0.5}\Big{(}\frac{2t^{2}||\phi||^{2}z_{0}^{-0.5}}{n(1-z_{0}^{-0.5})^{2}(1-a-\epsilon)^{2}(\ln\sqrt{z_{0}})^{2}}+\frac{t^{2}z_{0}^{-0.5n}||\phi||^{2}}{(1-z_{0}^{-0.5})(1-a-\epsilon)\ln\sqrt{z_{0}}}\Big{)}$
$\displaystyle\quad+O_{6921}\Big{(}\frac{|t|^{4}||\phi||^{4}}{n^{0.1}(1-z_{0}^{-0.5})^{4}(1-a-\epsilon)^{4}(\ln\sqrt{z_{0}})^{4}}\Big{)}+\sum_{i=1}^{n^{0.2}}O_{1}\Big{(}\frac{t^{2}\mathbb{E}|\mathbb{E}[X^{2}_{i}-\mathbb{E}X^{2}_{i}|\mathcal{F}_{i}]|}{n}\Big{)}.$

Now, by making use of (3.5), Theorem 2.1, and the fact that
$(\psi_{i}\circ\sigma^{i})_{i\geq 0}$ are reverse martingale differences, we
can continue the estimates as

$\displaystyle=\exp\Big{(}-\frac{t^{2}\Sigma^{2}}{2}\Big{)}+O_{6922}\Big{(}\frac{|t|^{4}||\phi||^{4}}{n^{0.1}(1-z_{0}^{-0.5})^{4}(1-a-\epsilon)^{4}(\ln\sqrt{z_{0}})^{4}}+\frac{t^{2}z_{0}^{-0.5n}||\phi||^{2}}{(1-z_{0}^{-0.5})(1-a-\epsilon)\ln\sqrt{z_{0}}}\Big{)}$
$\displaystyle\quad+\sum_{i=1}^{n^{0.2}}O_{1}\Big{(}\frac{\sum_{k\in
I_{i,1}}t^{2}\mathbb{E}|\mathbb{E}[\psi_{k}^{2}\circ\sigma^{k}-\mathbb{E}\psi^{2}_{k}\circ\sigma^{k}|\mathcal{F}_{i}]|}{n}\Big{)}$
$\displaystyle=\exp\Big{(}-\frac{t^{2}\Sigma^{2}}{2}\Big{)}+O_{6922}\Big{(}\frac{|t|^{4}||\phi||^{4}}{n^{0.1}(1-z_{0}^{-0.5})^{4}(1-a-\epsilon)^{4}(\ln\sqrt{z_{0}})^{4}}+\frac{t^{2}z_{0}^{-0.5n}||\phi||^{2}}{(1-z_{0}^{-0.5})(1-a-\epsilon)\ln\sqrt{z_{0}}}\Big{)}$
$\displaystyle\quad+\sum_{i=1}^{n^{0.2}}\sum_{k\in
I_{i,1}}O_{1}\Big{(}t^{2}||\psi_{k}^{2}||\frac{(\sqrt{z_{0}})^{-n^{0.2}}}{(1-a-\epsilon)\ln\sqrt{z_{0}}}\Big{)}$
$\displaystyle=\exp\Big{(}-\frac{t^{2}\Sigma^{2}}{2}\Big{)}+O_{6922}\Big{(}\frac{|t|^{4}||\phi||^{4}}{n^{0.1}(1-z_{0}^{-0.5})^{4}(1-a-\epsilon)^{4}(\ln\sqrt{z_{0}})^{4}}+\frac{t^{2}z_{0}^{-0.5n}||\phi||^{2}}{(1-z_{0}^{-0.5})(1-a-\epsilon)\ln\sqrt{z_{0}}}\Big{)}$
$\displaystyle\quad+O_{64}\Big{(}\frac{t^{2}n^{3}||\phi||^{2}e^{2||\phi_{p}||\theta(1-\theta)^{-1}}||\phi_{p}||^{2}(\sqrt{z_{0}})^{-n^{0.2}}}{(1-\theta)^{2}(1-a-\epsilon)\ln\sqrt{z_{0}}}\Big{)},$

which concludes a proof. ∎

######  A proof of Theorem 2.3.

We will apply the result of Azuma [1967] to $(\psi_{i}\circ\sigma^{i})_{i\geq
0}$:

$\mu_{\phi_{p}}\Big{(}\Big{|}\frac{\sum_{i\leq
n-1}\psi_{i}\circ\sigma^{i}}{n}\Big{|}\geq u/2\Big{)}\leq
2\exp\Big{(}-\frac{u^{2}n}{8\sup_{i}||\psi_{i}\circ\sigma^{i}||^{2}_{\infty}}\Big{)}.$

If
$n>\frac{2}{u}||\phi||\frac{z_{0}^{-0.5}}{(1-z_{0}^{-0.5})(1-a-\epsilon)\ln\sqrt{z_{0}}}$,
then, by (3.1), (3.2) and (3.3), we have that

$\displaystyle\mu_{\phi_{p}}\Big{(}\Big{|}\frac{\sum_{i\leq
n-1}\phi\circ\sigma^{i}}{n}\Big{|}\geq u\Big{)}$
$\displaystyle\leq\mu_{\phi_{p}}\Big{(}\Big{|}\frac{\sum_{i\leq
n-1}\psi_{i}\circ\sigma^{i}}{n}\Big{|}\geq
u/2\Big{)}+\mu_{\phi_{p}}\Big{(}\Big{|}\frac{H_{n}\circ\sigma^{n}}{n}\Big{|}\geq
u/2\Big{)}$ $\displaystyle=\mu_{\phi_{p}}\Big{(}\Big{|}\frac{\sum_{i\leq
n-1}\psi_{i}\circ\sigma^{i}}{n}\Big{|}\geq u/2\Big{)}$ $\displaystyle\leq
2\exp\Big{(}-\frac{u^{2}n}{8\sup_{i}||\psi_{i}\circ\sigma^{i}||^{2}_{\infty}}\Big{)}$
$\displaystyle\leq
2\exp\Big{(}-\frac{u^{2}n(1-z_{0}^{-0.5})^{2}(1-a-\epsilon)^{2}(\ln\sqrt{z_{0}})^{2}}{72||\phi||^{2}}\Big{)}.$

Hence, if $n\geq 1$, then

$\displaystyle\mu_{\phi_{p}}\Big{(}\Big{|}\frac{\sum_{i\leq
n-1}\phi\circ\sigma^{i}}{n}\Big{|}\geq u\Big{)}$ $\displaystyle\leq
2\exp\Big{(}\frac{u(1-z_{0}^{-0.5})(1-a-\epsilon)(\ln\sqrt{z_{0}})}{36||\phi||z_{0}^{0.5}}\Big{)}\exp\Big{(}-\frac{u^{2}n(1-z_{0}^{-0.5})^{2}(1-a-\epsilon)^{2}(\ln\sqrt{z_{0}})^{2}}{72||\phi||^{2}}\Big{)}.$

∎

###### A proof of Theorem 2.4.

Take now any $\delta>0$, and choose $u=n^{-(0.5-\delta)}$,

$\displaystyle\mu_{\phi_{p}}\Big{(}\Big{|}\frac{\sum_{i\leq
n-1}\phi\circ\sigma^{i}}{n}\Big{|}\geq n^{-(0.5-\delta)}\Big{)}$
$\displaystyle\leq
2\exp\Big{(}\frac{(1-z_{0}^{-0.5})(1-a-\epsilon)(\ln\sqrt{z_{0}})}{36||\phi||z_{0}^{0.5}}\Big{)}\exp\Big{(}-\frac{n^{2\delta}(1-z_{0}^{-0.5})^{2}(1-a-\epsilon)^{2}(\ln\sqrt{z_{0}})^{2}}{72||\phi||^{2}}\Big{)},$
$\displaystyle\mu_{\phi_{p}}\Big{(}\bigcup_{n\geq N}\Big{|}\frac{\sum_{i\leq
n-1}\phi\circ\sigma^{i}}{n}\Big{|}\geq n^{-(0.5-\delta)}\Big{)}$
$\displaystyle\leq
2\exp\Big{(}\frac{(1-z_{0}^{-0.5})(1-a-\epsilon)(\ln\sqrt{z_{0}})}{36||\phi||z_{0}^{0.5}}\Big{)}\sum_{n\geq
N}\exp\Big{(}-\frac{n^{2\delta}(1-z_{0}^{-0.5})^{2}(1-a-\epsilon)^{2}(\ln\sqrt{z_{0}})^{2}}{72||\phi||^{2}}\Big{)}.$

Let $N_{x,\delta}:=\sum_{N\geq 0}\mathbbm{1}_{\bigcup_{n\geq N}|\sum_{i\leq
n-1}\phi\circ\sigma^{i}(x)|\geq n^{0.5+\delta}}$. Then, by the Borel-Cantelli
lemma, for any $n\geq N_{x,\delta}$,

$\Big{|}\frac{\sum_{i\leq n-1}\phi\circ\sigma^{i}}{n}\Big{|}\leq
n^{-(0.5-\delta)},$

and $||N_{x,\delta}||_{1}$

$\displaystyle\leq
2\exp\Big{(}\frac{(1-z_{0}^{-0.5})(1-a-\epsilon)(\ln\sqrt{z_{0}})}{36||\phi||z_{0}^{0.5}}\Big{)}\sum_{N\geq
0}\sum_{n\geq
N}\exp\Big{(}-\frac{n^{2\delta}(1-z_{0}^{-0.5})^{2}(1-a-\epsilon)^{2}(\ln\sqrt{z_{0}})^{2}}{72||\phi||^{2}}\Big{)}$
$\displaystyle\leq
2\exp\Big{(}\frac{(1-z_{0}^{-0.5})(1-a-\epsilon)(\ln\sqrt{z_{0}})}{36||\phi||z_{0}^{0.5}}\Big{)}\sum_{n\geq
0}n\exp\Big{(}-\frac{n^{2\delta}(1-z_{0}^{-0.5})^{2}(1-a-\epsilon)^{2}(\ln\sqrt{z_{0}})^{2}}{72||\phi||^{2}}\Big{)},$

which concludes a proof. ∎

##  4 Applications

In this section we consider applications of the obtained results to some
uniformly hyperbolic linear maps $f\in SL(2d,\mathbb{Z})$ of $2d-$dim flat
torus ($d\geq 1$). These maps can be lifted to a topological Markov shift
$(\mathbb{A},\sigma,\mu_{\phi_{p}})$. We assume also that $f$ is a symmetric
matrix with the eigenvalues
$\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{d}>1>\lambda_{d+1}\geq\lambda_{d+2}\geq\cdots\geq\lambda_{2d}>0$.

Thanks to the linearity, the Radon–Nikodym derivative along unstable manifolds
(which are in this case linear sub-spaces) $\det D^{u}f=\exp(-\phi_{p})$ is
constant, and $||\phi_{p}||_{\infty}=|\ln\det D^{u}f|$. Since $f$ is
symmetric, then $||\phi_{p}||_{\infty}=|\ln\det
D^{u}f|=\sum_{i=1}^{d}\ln\lambda_{i}$. We choose
$||\phi_{p}||=\max\\{1,\sum_{i=1}^{d}\ln\lambda_{i}\\}$, and take a value of
$\theta$ in Definition 1 equal to $\lambda_{d}^{-1}$. Now let
$A_{d}=\bigoplus_{i=1}^{d}\begin{pmatrix}2&1\\\
1&1\end{pmatrix}:\bigoplus_{i=1}^{d}\mathbb{T}^{2}\to\bigoplus_{i=1}^{d}\mathbb{T}^{2}$
where $\bigoplus_{i=1}^{d}\mathbb{T}^{2}$ is a $2d$-dimensional torus.

######  Example.

Let $f=A_{d}B_{d}A_{d}$, where $B_{d}=\begin{pmatrix}1&\textbf{0}&1\\\
\textbf{0}&A_{d-1}&\textbf{0}\\\ 1&\textbf{0}&2\end{pmatrix}$. By making use
of the argument above, one can find $||\phi_{p}||$ and $\theta$. The
eigenvalues of $f$ are

$[(9+6\cos(2\pi j/d))\pm\sqrt{(9+6\cos(2\pi j/d))^{2}-4}]/2,$

$j=1,2,\cdots,d$. Then $\theta=(9-\sqrt{77})/2$,
$||\phi_{p}||=d\ln\frac{15+\sqrt{221}}{2}$. By Theorem 2.2,

$\Big{|}\mathbb{E}\exp\Big{(}it\frac{\sum_{0\leq i\leq
n-1}\phi\circ\sigma^{i}}{\sqrt{n}}\Big{)}-\exp\Big{(}-\frac{t^{2}\Sigma^{2}}{2}\Big{)}\Big{|}\leq
C_{t,n},$

where

$\displaystyle C_{t,n}=$ $\displaystyle
6922\Big{(}\frac{|t|^{4}||\phi||^{4}}{n^{0.1}(1-z_{0}^{-0.5})^{4}(1-a-\epsilon)^{4}(\ln\sqrt{z_{0}})^{4}}+\frac{t^{2}z_{0}^{-0.5n}||\phi||^{2}}{(1-z_{0}^{-0.5})(1-a-\epsilon)\ln\sqrt{z_{0}}}\Big{)}$
$\displaystyle+64\Big{(}\frac{t^{2}n^{3}||\phi||^{2}e^{2d\ln(\frac{15+\sqrt{221}}{2})\frac{9-\sqrt{77}}{\sqrt{77}-7}}d^{2}\ln^{2}\frac{15+\sqrt{221}}{2}(\sqrt{z_{0}})^{-n^{0.2}}}{(\frac{\sqrt{77}-7}{2})^{2}(1-a-\epsilon)\ln\sqrt{z_{0}}}\Big{)},$

and

$\epsilon\in(0,\min\\{1-a,\frac{\sqrt{77}-7}{2}\\}),\quad
a=1-\exp(-d\ln\frac{15+\sqrt{221}}{\sqrt{77}-7}),$
$z_{0}\in\Big{(}1,\min\Big{\\{}\Big{(}\frac{a+\epsilon/2}{a}\Big{)}^{U},\frac{9-\sqrt{77}+2\epsilon}{9-\sqrt{77}},\frac{2-2\epsilon}{9-\sqrt{77}},e^{2}\Big{\\}}\Big{)},\quad
U:=\frac{\log(1-\epsilon)}{\log[\frac{9-\sqrt{77}}{2}\epsilon(\frac{\sqrt{77}-7}{2}-\epsilon)]-\log(2||\phi_{p}||)}$

If $d=3$, then $a=0.9999$, $\epsilon=5\times 10^{-5}$, $z_{0}=1.00000000083$.
Therefore, the central limit theorem for $f$ holds, and

$\Big{|}\mathbb{E}\exp\Big{(}it\frac{\sum_{0\leq i\leq
n-1}\phi\circ\sigma^{i}}{\sqrt{n}}\Big{)}-\exp\Big{(}-\frac{t^{2}\Sigma^{2}}{2}\Big{)}\Big{|}\leq
357.15265\times 10^{56}\times t^{4}||\phi||^{4}n^{-0.1}.$

By Theorem 2.3, the large deviation estimate for $f$ is given via the
following relation

$\mu_{\phi_{p}}\Big{(}\Big{|}\frac{\sum_{i\leq
n-1}\phi\circ\sigma^{i}}{n}\Big{|}\geq u\Big{)}\leq 2\exp\Big{(}\frac{u\times
5.76388936\times 10^{-16}}{||\phi||}\Big{)}\exp\Big{(}-5.9800357\times
10^{-30}\times\frac{u^{2}n}{||\phi||^{2}}\Big{)}.$

By Theorem 2.4 in the law of large numbers we have for any $n\geq
N_{x,\delta},$

$\Big{|}\frac{\sum_{i\leq n}\phi\circ f^{i}(x)}{n}\Big{|}\leq
n^{-(0.5-\delta)},$

where $N_{x,\delta}:=\sum_{N\geq 0}\mathbbm{1}_{\bigcup_{n\geq N}|\sum_{i\leq
n-1}\phi\circ f^{i}(x)|\geq n^{0.5+\delta}}$ satisfies the relation

$||N_{x,\delta}||_{1}\leq 2\exp\Big{(}\frac{u\times 5.76388936\times
10^{-16}}{||\phi||}\Big{)}\sum_{n\geq 1}n\exp\Big{(}-5.9800357\times
10^{-30}\times\frac{n^{2\delta}}{||\phi||^{2}}\Big{)}<\infty.$

###### Acknowledgements.

We thank S. Vempala for the construction of the example in section 4 and
computing the eigenvalues. LB was partially supported by the NSF grant
DMS-2054659.

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