The Hausdorff dimension of multiply Xiong chaotic sets

TL;DR

This paper constructs a special type of chaotic set with full Hausdorff dimension in certain dynamical systems, demonstrating complex chaotic behavior with maximal fractal complexity.

Contribution

It introduces a multiply Xiong chaotic set with full Hausdorff dimension within multiply proximal cells for the full shift and Gauss systems, advancing understanding of chaos in these systems.

Findings

Constructed multiply Xiong chaotic set with full Hausdorff dimension

Set contained in multiply proximal cells for full shift and Gauss systems

Demonstrated existence of highly complex chaotic sets in these systems

Abstract

We construct a multiply Xiong chaotic set with full Hausdorff dimension everywhere that is contained in some multiply proximal cell for the full shift over finite symbols and the Gauss system respectively.

Equations

$$\underset{n\to \infty }{\mathrm{liminf}}\rho (T^{n}x,T^{n}y)=0\text{ and }\underset{n\to \infty }{\mathrm{limsup}}\rho (T^{n}x,T^{n}y)>0.$$

$$\underset{k\to \infty }{\lim }{T}^{q_k}(x)=g(x).$$

$$\underset{k\to \infty }{\lim }{T}^{j\cdot q_k}(x)=g_j(x),j=1,2,\dots ,d.$$

$$\overline{D}(A)=\underset{m\to \infty }{\mathrm{limsup}}\frac{1}{m}\#(A\cap [1,m]).$$

$$\mathcal{H}^{s}_{\delta}(A)=\inf\biggl{\{}\sum_{i\geq 1}\operatorname{diam}(U_{i})^{s}:A\subset\bigcup_{i\geq 1}U_{i}\text{ and }\operatorname{diam}(U_{i})<\delta\text{ for any }i\geq 1\biggr{\}}$$

$${\mathcal{H}}^s(A)=\underset{\delta \to 0}{\lim }{\mathcal{H}}_{\delta }^s(A)$$

$$\operatorname{dim_{H}}(A)=\biggl{\{}\begin{array}[]{ll}\inf\{s>0:\mathcal{H}^{s}(A)=0\},&\text{ if }\{s>0:\mathcal{H}^{s}(A)=0\}\neq\emptyset;\\ +\infty,&\text{ otherwise. }\end{array}\biggr{.}$$

$$\rho (x,y)=\{\begin{array}{ll}0,& \text{ if }x=y,\\ \frac{1}{N^k},& \text{ if }x\ne y\text{ and }k=\min \{n\in \mathbb{N}:x_n\ne y_n\}-1.\end{array}$$

$${\Phi }^{(1)}(x_1)=\underset{1\le i\le {l_1}^{l_1}}{⨆}{\phi }_{p_{i,1}^{(1)}}^{(1)}(x_1)U_{i,1}^{(1)}{\phi }_{p_{i,2}^{(1)}}^{(1)}(x_1)U_{i,2}^{(1)}\dots U_{i,l_1-1}^{(1)}{\phi }_{p_{i,l_1}^{(1)}}^{(1)}(x_1).$$

$${\Phi }^{(2)}(x_1{x}_2)={\Phi }^{(1)}(x_1)\bigsqcup \underset{1\le i\le {l_2}^{l_2}}{⨆}{\phi }_{p_{i,1}^{(2)}}^{(2)}(x_1{x}_2)U_{i,1}^{(2)}{\phi }_{p_{i,2}^{(2)}}^{(2)}(x_1{x}_2)U_{i,2}^{(2)}\dots U_{i,l_2-1}^{(2)}{\phi }_{p_{i,l_2}^{(2)}}^{(2)}(x_1{x}_2).$$

$${\Phi }^{(n)}(x_1{x}_2\cdots x_n)=$$

$$\displaystyle\sqcup\varphi_{p_{i,2}^{(n)}}^{(n)}(x_{1}x_{2}\dotsb x_{n})U_{i,2}^{(n)}\dotsc U_{i,l_{n}-1}^{(n)}\varphi_{p_{i,l_{n}}^{(n)}}^{(n)}(x_{1}x_{2}\dotsb x_{n})\Bigr{)}.$$

$${\Delta }_N^{(1)}(x)=V_{1,1}z[s_1+1]V_{1,2}z[1]{\Phi }^{(1)}(x_1).$$

$$Q_{1}=V_{1,1}V_{1,2}\sqcup\Biggl{(}\bigsqcup_{1\leq i\leq l_{1}^{l_{1}}}\bigsqcup_{1\leq j\leq l_{1}-1}U_{i,j}^{(1)}\Biggr{)}$$

$$\frac{\#\{j\le m:j\in D_1\}}{m}<\frac{2}{m}<\frac{2}{s_1+1}\le 1.$$

$$\frac{\#\{j\le m:j\in D_1\}}{m}<\frac{t_1}{2s_1+1}<1.$$

$${\Delta }_N^{(2)}(x)={\Delta }_N^{(1)}(x)V_{2,1}z[A_1^{(2)}+1,A_1^{(2)}+2]V_{2,2}z[A_1^{(2)}+1,A_1^{(2)}+2]V_{2,3}z[1,2]V_{2,4}z[1,2]{\Phi }^{(2)}(x_1{x}_2)$$

$$Q_{2}=Q_{1}\sqcup V_{2,1}V_{2,2}V_{2,3}V_{2,4}\sqcup\Biggl{(}\bigsqcup_{1\leq i\leq l_{2}^{l_{2}}}\bigsqcup_{1\leq j\leq l_{2}-1}U_{i,j}^{(2)}\Biggr{)}$$

$$\frac{\#\{j\le m:j\in D_1\cup D_2\}}{m}<\frac{t_1}{2s_1+1+c_1}<1.$$

$$\frac{\#\{j\le m:j\in D_1\cup D_2\}}{m}<\frac{t_1+8}{A_1^{(2)}}<\frac{1}{2}.$$

$$\frac{\#\{j\le m:j\in D_1\cup D_2\}}{m}<\frac{t_1+t_2}{A_1^{(2)}+2+A_2^{(2)}+2+A_3^{(2)}+2+A_4^{(2)}+2}<\frac{1}{2}.$$

$${\Delta }_N^{(n)}(x)=$$

$$\bigsqcup V_{n,n}z[A_1^{(n)}+1,A_1^{(n)}+n]\bigsqcup V_{n,n+1}z[1,n]V_{n,n+2}z[1,n]\bigsqcup \cdots \bigsqcup V_{n,2n}z[1,n]$$

$$\bigsqcup {\Phi }^{(n)}(x_1{x}_2\cdots x_n)$$

$$Q_{n}=Q_{n-1}\sqcup\bigsqcup_{j=1}^{2n}V_{n,j}\sqcup\Biggl{(}\bigsqcup_{1\leq i\leq l_{n}^{l_{n}}}\bigsqcup_{1\leq j\leq l_{n}-1}U_{i,j}^{(n)}\Biggr{)}$$

$$\frac{\#\{j\le m:j\in {\bigcup }_{i=1}^n{D}_i\}}{m}\le \frac{1}{n-1}.$$

$$\frac{\#\{j\le m:j\in {\bigcup }_{i=1}^n{D}_i\}\}}{m}<\frac{t_{n-1}+2n^2}{A_1^{(n)}}<\frac{1}{n}.$$

$$\frac{\#\{j\le m:j\in {\bigcup }_{i=1}^n{D}_i\}\}}{m}<\frac{t_{n-1}+t_n}{A_1^{(n)}+A_2^{(n)}+\cdots +A_{2n}^{(n)}+2n^2}<\frac{1}{n}.$$

$$\frac{\#({\bigcup }_{i=1}^{\infty }{D}_i\cap [1,m])}{m}<\frac{\#({\bigcup }_{i=1}^{n+1}{D}_i\cap [1,m])}{m}<\frac{1}{n}\le \frac{1}{k}.$$

$${\psi }_{k,j}(x)=\max \{0\le i\le k:$$

$$x[1,k]\cap E\subset g_j^{-1}([Y_{i,j}])\}.$$

$${\phi }_{p_{i,1}^{(k)}}^{(k)}(x_1{x}_2\cdots x_k)[1,{\psi }_{k,1}(x)]=Y_{k,1}(x),$$

Full text

1. Introduction

By a topological dynamical system, we mean a pair $(X,T)$, where $X$ is a metric space and $T$ is a continuous self-map of $X$. A compatible metric on $X$ is denoted by $\rho$. A subset $S$ of $X$ is said to be scrambled if any two distinct points $x,y\in S$ satisfy

[TABLE]

We say that a topological dynamical system $(X,T)$ is Li-Yorke chaotic if there exists an uncountable scrambled subset of $X$. As Li-Yorke chaos only requires an uncountable scrambled set, it is natural to ask how “large” a scrambled set can be. There are two approaches about the size of a set: topological and measure-theoretic. In [BHS08], Blanchard, Huang and Snoha studied the topological size of scrambled sets extensively, see also the references therein. In [BL99], Balibrea and Jiménez López surveyed the Lebesgue measure of scrambled sets for continuous maps on the interval till then. In [B09], Bruin and Jiménez López studied the Lebesgue measure of scrambled sets for $C^{2}$ and $C^{3}$ multimodal interval maps $f$ with non-flat critical points. Recently, in [Li2017] Liu and Li constructed a scrambled set with full Hausdorff dimension for the Gauss system.

In the study of the complexity of weakly mixing system, Xiong introduced a kind of chaos in [XX91] and [X91], which is called Xiong chaos nowadays. We say that a subset $E$ of $X$ containing at least two points is Xiong chaotic if for any subset $A$ of $E$ and any continuous function $g\colon A\to X$, there exists an increasing sequence $\{q_{k}\}$ of positive integers such that for every $x\in A$,

[TABLE]

It is clear that every Xiong chaotic set is scrambled. It is shown in [X91] that a non-trivial topological dynamical system is weakly mixing if and only if it has a $F_{\sigma}$, $c$-dense, Xiong chaotic set. In [X95], Xiong constructed a Xiong chaotic set with full Hausdorff dimension everywhere in the full shift over finite symbols. Wu and Tan generalized this result to the full shift over countable symbols in [WT07].

In [BH08], Blanchard and Huang introduced a local version of weak mixing and showed that if a topological dynamical system has positive topological entropy, then it has “many” weakly mixing sets and then “local” Xiong chaotic sets. Recently, Huang et. al generalized this result to $\Delta$-weakly mixing sets [HLYZ17]. In fact, inspired by the proof of Theorem A in [HLYZ17], we can proposal the following concept of multiply Xiong chaotic set. We say that a subset $E$ of $X$ containing at least two points is multiply Xiong chaotic if for any $d\in\mathbb{N}$, any subset $A$ of $E$, and continuous functions $g_{j}\colon B\to X$ for $j=1,2,\dotsc,d$, there exists an increasing sequence $\{q_{k}\}$ of positive integers such that for every $x\in A$,

[TABLE]

With this terminology, combining with Theorem A and Proposition 3.2 in [HLYZ17], we have that a non-trivial topological dynamical system is $\Delta$-transitive if and only if it has a dense, $\sigma$-Cantor, multiply Xiong chaotic set. In addition, the author in [Liu2019] showed that positive topological entropy implies the existence of $\Delta$-weakly mixing subsets for finitely generated torsion-free discrete nilpotent group actions.

The main aim of this paper is to study the Hausdorff dimension of multiply Xiong chaotic sets in the full shift and the Gauss system. In fact, we will study chaotic sets in the multiply proximal cell of a point.

Recall that a pair $(x,y)\in X\times X$ is proximal if $\liminf_{n\to\infty}\rho(T^{n}x,T^{n}y)=0$ and the proximal cell of a point $x\in X$ is defined by $Prox(x,T)=\{y\in X\colon(x,y)\textrm{ is proximal}\}$. The structure of the proximal cell plays an important role in a topological dynamical system. Many results concerning the structure of proximal cell have been studied. For example, Auslander and Ellis proved that every proximal cell contains a minimal point for a compact system [Au60]. Moreover, in a weakly mixing system every proximal cell is residual [AK03]. In [HW04], Huang, Shao and Ye proved an equivalent statement of an $\mathcal{F}$-mixing system by the dense $G_{\delta}$ structure of $Prox(x,T)$ for any $x\in X$ and they gave a detailed description of the proximal cells for $\mathcal{F}$-mixing systems where $\mathcal{F}$ is a Furstenberg family satisfying some special properties.

We generalize the proximal to multiply proximal as follows. A pair $(x,y)\in X\times X$ is called multiply proximal if for any $d\in\mathbb{N}$, the pair $(x,y)$ satisfies $\liminf_{n\to\infty}\max_{1\leq j\leq d}$ $\rho(T^{j\cdot n}(x),T^{j\cdot n}(y))=0$ and the multiply proximal cell of a point $x\in X$ is denoted by $MProx(x,T)=\{y\in X\colon(x,y)\textrm{ is multiply proximal}\}$. It is clear that $MProx(x,T)$ is a subset of $Prox(x,T)$. We will study the multiply Xiong chaos in multiply proximal cells for some special systems. Now we are ready to state the main results of this paper as follows:

Theorem 1.1**.**

Let $N\geq 2$ be a positive integer. In the full shift $(\Sigma_{N},\sigma)$ over $N$-symbols, for every $z\in\Sigma_{N}$, the multiply proximal cell of $z$ contains a multiply Xiong chaotic sets with full Hausdorff dimension everywhere.

The Gauss map $T\colon[0,1)\to[0,1)$ is defined by T(x)=\frac{1}{x}-\bigl{\lfloor}\frac{1}{x}\bigr{\rfloor} for $x\in(0,1)$ and $T(0)=0$, where $\lfloor z\rfloor$ denote the greatest integer less than or equal to $z$. The restriction of $T$ on $[0,1)\setminus\mathbb{Q}$ is called the Gauss system.

Theorem 1.2**.**

In the Gauss system $([0,1)\setminus\mathbb{Q},T)$, for every $z$ in $[0,1)\setminus\mathbb{Q}$, the multiply proximal cell of $z$ contains a multiply Xiong chaotic sets with full Hausdorff dimension everywhere.

The paper is organized as follows. In Section 2, we introduce some preliminaries. Section 3 is devoted to proving Theorem  1.1. In Section 4, Theorem 1.2 is proved and we discuss the properties of the scrambled set for the Gauss system.

2. Preliminaries

In this section, we present some basic notations, definitions and results that will be used later.

2.1. Upper density

Denote $\mathbb{N}$ the set of positive integers. For a finite subset $F$ of $\mathbb{N}$, denote the cardinality of $F$ by $\#(F)$. For a subset $A$ of $\mathbb{N}$, the upper density of $A$ is defined as

[TABLE]

For a strictly increasing sequence in $\mathbb{N}$, we can view it as an infinite subset of $\mathbb{N}$ and define the upper density of this sequence as the one of the infinite subset.

2.2. Hausdorff dimension

In a metric space $(X,\rho)$, for a subset $A$ of $X$, a real number $\delta>0$, and $s\geq 0$, define

[TABLE]

where $\operatorname{diam}(\cdot)$ denotes the diameter of a set. The $s$-dimension Hausdorff measure of $A$ is given by

[TABLE]

and the Hausdorff dimension of $A$ is

[TABLE]

The basic knowledge about Hausdorff dimension can be found in [Fa2003], which we refer the reader to. We say that a subset $C$ of $X$ has full Hausdorff dimension everywhere if $\operatorname{dim_{H}}(C\cap U)=\operatorname{dim_{H}}(X)$ for any non-empty open subset $U$ of $X$ .

Let $\alpha$ be a positive real number. We say that a map $f:X\to\mathbb{R}$ satisfies the locally $\alpha$-Hölder condition if there exist a real number $r>0$ and a constant $c>0$ such that |f(x)-f(y)|\leq c\bigl{(}\rho(x,y)\bigr{)}^{\alpha} holds for any $x,y\in X$ with $\rho(x,y)<r$.

The following well known lemma can be easily deduced from the definitions of Hausdorff dimension and the locally $\alpha$-Hölder condition.

Lemma 2.1**.**

Let $(X,\rho)$ be a metric space and $s,\alpha>0$ be real numbers. If a map $f:X\to\mathbb{R}$ satisfies the locally $\alpha$-Hölder condition, then $\mathcal{H}^{s}(f(X))\leq c^{s}\mathcal{H}^{s\cdot\alpha}(X)$, where $c$ is the constant in the definition of the locally $\alpha$-Hölder condition. Moreover, $\alpha\operatorname{dim_{H}}(f(X))\leq\operatorname{dim_{H}}(X)$.

2.3. Ergodic theory

For a probability space $(X,\mathcal{B},\mu)$, a transformation $T\colon X\to X$ is called measure-preserving if $T^{-1}(\mathcal{B})\subset\mathcal{B}$ and $\mu(A)=\mu(T^{-1}(A))$ for any $A$ in the $\sigma$-algebra $\mathcal{B}$. In this case, the quadruple $(X,\mathcal{B},\mu,T)$ is called a measure-preserving system. A measure-preserving transformation $(X,\mathcal{B},\mu,T)$ is called ergodic if the only members $A$ of $\mathcal{B}$ with $T^{-1}(A)=A$ satisfies $\mu(A)=0$ or $\mu(A)=1$, weakly mixing if $(X\times X,\mathcal{B}\times\mathcal{B},\mu\times\mu,T\times T)$ is ergodic, and strongly mixing if $\lim\limits_{n\to\infty}m(T^{-n}(A)\cap B)=m(A)m(B)$ for any $A,B\in\mathcal{B}$. It is obviously that every strongly mixing transformation is weakly mixing and every weakly mixing transformation is ergodic.

We say that a measure-preserving system $(X,\mathcal{B},\mu,T)$ is exact if for each $B$ in the tail $\sigma$-algebra $\bigcap_{n\in\mathbb{N}}T^{-n}(\mathcal{B})$, either $\mu(B)$ or $\mu(X\setminus B)$ is zero. The exactness was introduced by Rokhlin in [R61] and he obtained the following result.

Proposition 2.2** ([R61]).**

1. (1)

If a measure-preserving system $(X,\mathcal{B},m,T)$ is exact, then it is strongly mixing. 2. (2)

A measure-preserving system $(X,\mathcal{B},m,T)$ is exact if and only if $\lim_{n\to\infty}m(T^{n}A)=1$ for every $A\in\mathcal{B}$ with $m(A)>0$ and $T^{n}A\in\mathcal{B}$ for every $n\geq 1$.

3. multiply Xiong chaotic sets in full

shift over finite symbols

In this section, we construct a multiply Xiong chaotic set with full Hausdorff dimension everywhere for the full shift over finite symbols, which proves Theorem 1.1.

3.1. The full shift over finite symbols

Let $N$ be an integer with $N\geq 2$. We endow the finite set $\{1,2,\dotsc,N\}$ with the discrete topology and the product space $\Sigma_{N}=\prod_{n=1}^{\infty}\{1,2,\dotsc,N\}$ is compact and metrizable. A compatible metric $\rho$ on $\Sigma_{N}$ can be defined as follow. For any $x=x_{1}x_{2}\dotsb$ and $y=y_{1}y_{2}\dotsb\in\Sigma_{N}$,

[TABLE]

The shift map $\sigma\colon\Sigma_{N}\to\Sigma_{N}$ is defined as $\sigma(x)=x_{2}x_{3}\dotsb$ for any $x=x_{1}x_{2}\dotsb\in\Sigma_{N}$. It is clear that $\sigma$ is continuous. The dynamical system $(\Sigma_{N},\sigma)$ is called the full shift over $N$ symbols.

Let $n\geq 1$. Each element in $\{1,2,\dotsc,N\}^{n}$ is called a word with length $n$ and denote the set consisting of all the words by $\mathcal{N}^{*}$. For a point $x=x_{1}x_{2}\dotsb\in\Sigma_{N}$, denote its prefix with length $n$ by $x[1,n]=x_{1}x_{2}\dotsb x_{n}\in\{1,2,\dotsc,N\}^{n}$. We also use $x[n]$ to represent the number at the $n$-position of $x$ for any $n\geq 1$. We say that the number $n$ is the position of the word $x[n,m]$ appeared in the point $x$ for any $m\geq n$. The cylinder generated by the word $x_{1}x_{2}\dotsb x_{n}$ is the set consisting of the points with the same word $x_{1}x_{2}\dotsb x_{n}$ as its prefix and denote it by $[x_{1}x_{2}\dotsb x_{n}]=\{y\in\Sigma_{N}:y_{i}=x_{i}\text{ for }i=1,2,\dotsc,n\}$. Clearly, a cylinder is both open and closed. For $u=u_{1}u_{2}\dotsb u_{n}$ and $v=v_{1}v_{2}\dotsb v_{m}$, denote by $uv$ or $u\sqcup v$ the concatenation of $u$ and $v$, that is $uv=u\sqcup v=u_{1}u_{2}\dotsb u_{n}v_{1}v_{2}\dotsb v_{m}$. Since $u\sqcup v$ may be different from $v\sqcup u$, we require that the symbol “$\bigsqcup_{1\leq i\leq n}u_{i}$ ” means $u_{1}\sqcup u_{2}\sqcup\dotsb\sqcup u_{n}$. It is not hard to show that with the metric $\rho$, any cylinder has full Hausdorff dimension, which is equal to $1$.

The following result was proved in [X95, Lemma 3], which is a key estimation of the Hausdorff dimension of a subset of $\Sigma_{N}$.

Lemma 3.1**.**

Let $A=\{a_{n}\}_{n=1}^{\infty}$ be a sequence of strictly increasing positive numbers. Define a map $\Gamma_{A}:\Sigma_{N}\to\Sigma_{N}$, $\Gamma_{A}(x)=x_{1}x_{2}\dotsc x_{a_{1}-1}x_{a_{1}+1}\dotsc x_{a_{n}-1}x_{a_{n}+1}\dotsc$. Let $Y$ be a subset of $\Sigma_{N}$. If the upper density of $A$ is $\lambda$ and $\mathcal{H}^{1}(\Gamma_{A}(Y))>0$, then $\operatorname{dim_{H}}(Y)\geq 1-\lambda$.

3.2. Proof of Theorem 1.1

Now we construct a multiply Xiong chaos set step by step. Note that the main idea comes from [X95], but we should amend the method to multiply Xiong chaotic case. We first define a map $\Delta_{N}\colon\Sigma_{N}\to\Sigma_{N}$ to construct a subset, then we show that this subset meets the requirement.

Fix a point $z=z_{1}z_{2}\dotsc\in\Sigma_{N}$. For each $n\in\mathbb{N}$, let $l_{n}=(N^{n})^{N^{n}}$. We list all the self-maps on $\{1,2,\dotsc,N\}^{n}$ as $\varphi_{1}^{(n)},\varphi_{2}^{(n)},\dotsc,\varphi_{l_{n}}^{(n)}$. Fix a point $x=x_{1}x_{2}\dotsb$ in $\Sigma_{N}$. To define $\Delta_{N}(x)$, we first use these maps $\varphi_{i}^{(n)}$ to construct a series of words functioning as the chaotic part. For any integer $1\leq i\leq{l_{n}}^{l_{n}}$ and $1\leq j\leq l_{n}-1$, we choose a word $U_{i,j}^{(n)}$ in $\mathcal{N}^{*}$ that will be specialized later. We list the elements in $\{1,2,\dotsc,l_{n}\}^{l_{n}}$ as \mathbf{p}_{i}^{(n)}=\bigl{(}p_{i,1}^{(n)},p_{i,2}^{(n)},\dotsb,p_{i,l_{n}}^{(n)}\bigr{)}, $1\leq i\leq{l_{n}}^{l_{n}}$.

First, we deal with the case when $k=1$. Define a word as follow,

[TABLE]

Second, when $k=2$, define a word

[TABLE]

Continuing this process, for each $n\in\mathbb{N}$, we can define a word

[TABLE]

Denote the length of $\Phi^{(n)}(x_{1}x_{2}\dotsb x_{n})$ by $c_{n}$.

Next, we start to define another series of words in order to meet the requirement of the proximal part. Set $s_{n}=(l_{n})^{l_{n}+1}$ for any $n\geq 1$. When $k=1$, define a word

[TABLE]

where $V_{1,1}$ and $V_{1,2}$ are two words that will be defined later. We determine the word $\Delta_{N}^{(1)}(x)$ by the following rules:

1. (1)

the length of $V_{1,1}$ is equal to $s_{1}$; 2. (2)

the length of $V_{1,2}$ is equal to $s_{1}-1$; 3. (3)

we choose a proper length of each $U_{i,j}^{(1)}$ such that $r_{i,j}^{(1)}-1=j(r_{i,1}^{(1)}-1)$ for $1\leq i\leq{l_{1}}^{l_{1}}$ and $1\leq j\leq l_{1}-1$, where $r_{i,j}^{(1)}=r_{i,j}^{(1)}(s_{1})$ is the position of the word $\varphi_{p_{i,j}^{(1)}}^{(1)}(x_{1})$ appeared in $\Delta_{N}^{(1)}(x)$. 4. (4)

we specialize the words $V_{1,j}$ and $U_{i,j}^{(1)}$ such that the word

[TABLE]

is an initial segment of $x$, that is $Q_{1}=x[1,q_{1}]$, where $q_{1}$ is the length of $Q_{1}$.

Let $t_{1}=s_{1}+2$. Denote by $D_{1}=\{a_{1}^{(1)}<a_{2}^{(1)}<\dotsb<a_{t_{1}}^{(1)}\}$ the set of the positions of $z[s_{1}+1]$, $z[1]$, and $\varphi_{p_{i,1}^{(1)}}^{(1)}(x_{1})$, $\varphi_{p_{i,2}^{(1)}}^{(1)}(x_{1})$, $\dotsc$, $\varphi_{p_{i,l_{1}}^{(1)}}^{(1)}(x_{1})$ for $1\leq i\leq{l_{1}}^{l_{1}}$ appeared in $\Delta_{N}^{(1)}(x)$. Note that the position of $z[s_{1}+1]$ appeared in $\Delta_{N}^{(1)}(x_{1})$ is $s_{1}+1$ and the position of $z[1]$ appeared in $\Delta_{N}^{(1)}(x_{1})$ is $2s_{1}+1$. We estimate the density of the set $D_{1}$ as follow.

1. (i)

If an integer $m$ satisfies $s_{1}+1\leq m\leq 2s_{1}+1$, then

[TABLE] 2. (ii)

If $2s_{1}+1<m\leq c_{1}+2s_{1}+1$, then

[TABLE]

When $k=2$, define a word

[TABLE]

where $A_{i}^{(2)}$ is a positive integer and $V_{2,i}$ is a word for $1\leq i\leq 4$, which will be defined later. We determine the word $\Delta_{N}^{(2)}(x)$ by the following rules:

1. (1)

$A_{1}^{(2)}>2(t_{1}+8)$ and $A_{4}^{(2)}>t_{1}+t_{2}-3$, where $t_{2}=2s_{2}+8$; 2. (2)

the length of $\Delta_{N}^{(1)}(x)V_{2,1}$ is equal to $A_{1}^{(2)}$ and the length of $V_{2,i}$ is $A_{i}^{(2)}$ for $2\leq i\leq 4$; 3. (3)

$2+A_{2}^{(2)}=A_{1}^{(2)}$ and $A_{1}^{(2)}+2+A_{2}^{(2)}+2+A_{3}^{(2)}=2+A_{4}^{(2)}$; 4. (4)

we choose a proper length of each $U_{i,j}^{(2)}$ such that $r_{i,j}^{(2)}-1=j(r_{i,1}^{(2)}-1)$ for $1\leq i\leq{l_{2}}^{l_{2}}$ and $1\leq j\leq l_{2}-1$, where $r_{i,j}^{(2)}=r_{i,j}^{(2)}(A_{1}^{(2)},A_{4}^{(2)})$ is the position of the word $\varphi_{p_{i,j}^{(2)}}^{(2)}(x_{1}x_{2})$ appeared in $\Delta_{N}^{(2)}(x)$; 5. (5)

we specialize the words $V_{2,j}$ and $U_{i,j}^{(2)}$ such that the word

[TABLE]

is an initial segment of $x$.

We denote by $D_{2}=\{a_{1}^{(2)}<a_{2}^{(2)}<\dotsb<a_{t_{2}}^{(2)}\}$ the set of the positions of $z[A_{1}^{(2)}+1]$, $z[A_{1}^{(2)}+2]$, $z[1]$, $z[2]$ $\varphi_{p_{i,1}^{(2)}}^{(2)}(x_{1}x_{2})[1]$, $\varphi_{p_{i,1}^{(2)}}^{(2)}(x_{1}x_{2})[2]$ $\varphi_{p_{i,2}^{(2)}}^{(2)}(x_{1}x_{2})[1]$, $\varphi_{p_{i,2}^{(2)}}^{(2)}(x_{1}x_{2})[2]$, $\dotsc$ , $\varphi_{p_{i,l_{2}}^{(2)}}^{(2)}(x_{1}x_{2})[1]$, $\varphi_{p_{i,l_{2}}^{(2)}}^{(2)}(x_{1}x_{2})[2]$ for $1\leq i\leq l_{2}^{l_{2}}$ appeared in $\Delta_{N}^{(2)}(x)$. We estimate the density of set $D_{1}\cup D_{2}$ as follow.

1. (i)

If $c_{1}+2s_{1}+1<m\leq A_{1}^{(2)}$, then

[TABLE] 2. (ii)

If $A_{1}^{(2)}<m\leq A_{1}^{(2)}+2+A_{2}^{(2)}+2+A_{3}^{(2)}+2+A_{4}^{(2)}+2$, then

[TABLE] 3. (iii)

If $A_{1}^{(2)}+2+A_{2}^{(2)}+2+A_{3}^{(2)}+2+A_{4}^{(2)}+2<m\leq A_{1}^{(2)}+2+A_{2}^{(2)}+2+A_{3}^{(2)}+2+A_{4}^{(2)}+c_{2}$, then

[TABLE]

Assume that we have completed the construction in the case $k=n-1$. We carry on the case $k=n$ as follows. Define a word

[TABLE]

where $A_{i}^{(n)}$ is a positive integer and $V_{n,i}$ is a word for $1\leq i\leq 2n$, which will be defined later. We determine the word $\Delta_{N}^{(n)}(x)$ by the following rules:

1. (R1)

$A_{1}^{(n)}>n(t_{n-1}+2n^{2})$ and $A_{2n}^{(n)}>t_{n-1}+t_{n}-(n+1)$, where $t_{n}=n\cdot s_{n}+2n^{2}$; 2. (R2)

the length of $\Delta_{N}^{(1)}(x)V_{n,1}$ is equal to $A_{1}^{(n)}$ and the length of $V_{n,i}$ is $A_{i}^{(n)}$ for $2\leq i\leq 2n$; 3. (R3)

$A_{1}^{(n)}=n+A_{2}^{(n)}=\dotsb=n+A_{n}^{(n)}$ and $A_{1}^{(n)}+A_{2}^{(n)}+\dotsb+A_{n}^{(n)}+n^{2}+A_{n+1}^{(n)}=n+A_{n+2}^{(n)}=n+A_{n+3}^{(n)}=\dotsb=n+A_{2n}^{(n)}$; 4. (R4)

we choose a proper length of each $U_{i,j}^{(n)}$such that $r_{i,j}^{(n)}-1=j(r_{i,1}^{(n)}-1)$ for $1\leq i\leq{l_{n}}^{l_{n}}$ and $1\leq j\leq l_{n}-1$, where $r_{i,j}^{(n)}=r_{i,j}^{(n)}(A_{1}^{(n)},A_{2n}^{(n)})$ is the position of the word $\varphi_{p_{i,j}^{(n)}}^{(n)}(x_{1}x_{2}\dotsb x_{n})$ appeared in $\Delta_{N}^{(n)}(x)$; 5. (R5)

we specialize the words $V_{n,j}$ and $U_{i,j}^{(n)}$ such that the word

[TABLE]

is an initial segment of $x$.

We denote by $D_{n}=\{a_{1}^{(n)}<a_{2}^{(n)}<\dotsb<a_{t_{n}}^{(n)}\}$ the set of the positions of $z[A_{1}^{(n)}+1]$, $z[A_{1}^{(n)}+2]$, $\dotsc$, $z[A_{1}^{(n)}+n]$, $z[1]$, $z[2]$,$\dotsc$, $z[n]$ and $\varphi_{p_{i,1}^{(n)}}^{(n)}(x_{1}x_{2}\dotsb x_{n})[1]$, $\varphi_{p_{i,1}^{(n)}}^{(n)}(x_{1}x_{2}\dotsb x_{n})[2]$, $\dotsc$, $\varphi_{p_{i,1}^{(n)}}^{(n)}(x_{1}x_{2}\dotsb x_{n})[n]$, $\varphi_{p_{i,2}^{(n)}}^{(n)}(x_{1}x_{2}\dotsb x_{n})[1]$, $\varphi_{p_{i,2}^{(n)}}^{(n)}(x_{1}x_{2}\dotsb x_{n})[2]$, $\dotsc$, $\varphi_{p_{i,2}^{(n)}}^{(n)}(x_{1}x_{2}\dotsb x_{n})[n]$, $\dotsc$, $\varphi_{p_{i,l_{n}}^{(n)}}^{(n)}(x_{1}x_{2}\dotsb x_{n})[1]$, $\varphi_{p_{i,l_{n}}^{(n)}}^{(n)}(x_{1}x_{2}\dotsb x_{n})[2]$, $\dotsc$, $\varphi_{p_{i,l_{n}}^{(n)}}^{(n)}(x_{1}x_{2}\dotsb x_{n})[n]$ for $1\leq i\leq{l_{n}}^{l_{n}}$ appeared in $\Delta_{N}^{(n)}(x)$. We estimate the density of set $\bigcup_{i=1}^{n}D_{i}$ as follow.

1. (i)

If $1\leq m\leq A_{1}^{(n)}$, then

[TABLE] 2. (ii)

If $A_{1}^{(n)}<m\leq A_{1}^{(n)}+A_{2}^{(n)}+\dotsb+A_{2n}^{(n)}+2n^{2}$, then

[TABLE] 3. (iii)

If $A_{1}^{(n)}+A_{2}^{(n)}+\dotsb+A_{2n}^{(n)}+2n^{2}<m\leq A_{1}^{(n)}+A_{2}^{(n)}+\dotsb+A_{2n}^{(n)}+2n^{2}+c_{n}$, then

[TABLE]

By induction the sequence $\{\Delta_{N}^{(n)}(x)\}_{n=1}^{\infty}$ is defined, now we define $\Delta_{N}(x)=\lim_{n\to\infty}\Delta_{N}^{(n)}(x)0^{\infty}$. It is easy to see that $\Delta_{N}$ is injective and continuous.

For any $n\geq 1$ and $v=v_{1}v_{2}\dotsb v_{n}\in\{1,2,\dotsc,N\}^{n}$, define a map $\theta_{v}:\Sigma_{N}\to\Sigma_{N}$ by $\theta_{v}(x)=v_{1}v_{2}\dotsb v_{n}x_{n+1}\cdots$ for any $x=x_{1}x_{2}\dotsb\in\Sigma_{N}$. It is clear that $\theta_{v}$ is a continuous and open map. Numerate the countable set $\bigcup_{k=1}^{\infty}\{1,2,\dotsc,N\}^{k}$ as $\{w_{n}:n\geq 1\}$. For any $n\geq 1$, set $B_{n}=[222\dotsb 21]\in\{1,2,\dotsc,N\}^{n+1}$ and $C_{n}=\theta_{w_{n}}\circ\Delta_{N}(B_{n})$. Then we make the countable union as $C=\bigcup_{n=1}^{\infty}C_{n}$ and $B=\bigcup_{n=1}^{\infty}\Delta_{N}(B_{n})$. Based on the fact that $\Delta_{N}(B_{n})$ is pairwise disjoint closed for any $n\geq 1$ (because both $\{B_{n}\}$ and $\{C_{n}\}$ are two sequences of disjoint closed subsets), we can define a map $\xi:B\to C$ such that $\xi|_{\Delta_{N}(B_{n})}=\theta_{w_{n}}|_{\Delta_{N}(B_{n})}$ for any $n\geq 1$. Note that $\Delta_{N}(B_{n})$ is a closed subset of $B$ for any $n\geq 1$, since $\Delta_{N}(B_{n})$ is a countable intersection of closed subsets of $\Sigma_{N}$. And $\theta_{w_{n}}$ is continuous for any $n\geq 1$, the map $\xi$ is continuous.

We now turn to show that $C$ is contained in the multiply proximal cell of the point $z$. For any $d\in\mathbb{N}$ and $y\in C$, there exists $n\in\mathbb{N}$ such that $y\in\theta_{w_{n}}\circ\Delta_{N}(B_{n})$. According to (R2) and (R3), we know that $\lim_{k\to\infty}\rho(\sigma^{j\cdot A_{1}^{(k)}}(z),\sigma^{j\cdot A_{1}^{(k)}}(y))=0$ for $j=1,2,\dotsc,d$. Hence, $C$ is contained in the multiply proximal cell of $z$. Furthermore, we can also obtain that $\lim_{k\to\infty}\operatorname{diam}(\{z\}\cup\sigma^{k+A_{2k}^{(k)}}(C))=0$.

Next, we show that $C$ has full Hausdorff dimension everywhere. First, it is not hard to show that $\operatorname{dim_{H}}(\Sigma_{N})=\operatorname{dim_{H}}(B_{n})=1$ for any $n\geq 1$. Second, let $D=\bigcup_{i=1}^{\infty}D_{i}$. By (R5) it is easy to see that $\Gamma_{D}\circ\Delta_{N}$ is identity. Third, we claim that the density of the set $\bigcup_{i=1}^{\infty}D_{i}$ is zero. The reason is that for any $\frac{1}{k}$ with $k$ being an integer larger than $2$, there exists an integer $A_{1}^{(k)}$ such that for any $m>A_{1}^{(k)}$ we can find some integer $n\geq k$ with $A_{1}^{(n)}<m\leq A_{1}^{(n+1)}$ such that

[TABLE]

Fourth, by Lemma 3.1, we know that $\operatorname{dim_{H}}(\Delta_{N}(B_{n}))=1$ because $\mathcal{H}^{1}(\Gamma_{D}\circ\Delta_{N}(B_{n}))=\mathcal{H}^{1}(B_{n})>0$ and the density of $D$ is zero. As the map $\theta_{w_{n}}$ only changes the prefix of the points in $B_{n}$ with length $n$, it turns out that $\operatorname{dim_{H}}(\theta_{w_{n}}(\Delta_{N}(B_{n})))=\operatorname{dim_{H}}(\Delta_{N}(B_{n}))=1$ for any $n\geq 1$. At last, for any non-empty open subset $U$ of $\Sigma_{N}$, there exists some $w_{n}$ such that $[w_{n}]\subset U$. Since $C_{n}=\theta_{w_{n}}\circ\Delta_{N}(B_{n})\subset U$, it is clear that $\operatorname{dim_{H}}(C\cap U)\geq\operatorname{dim_{H}}(C_{n}\cap U)=\operatorname{dim_{H}}(C_{n})=\operatorname{dim_{H}}(\theta_{w_{n}}(\Delta_{N}(B_{n})))=1$.

Finally, it remains to show that $C$ is multiply Xiong chaotic. As $\xi\colon B\to C$ is continuous, it is sufficient to show that $B$ is a multiply Xiong chaotic set. Let $E$ be a subset of $B$ and $g_{j}:E\to\Sigma_{N}$ be a continuous map for $j=1,2,\dotsc,d$. For any $x\in E$, any $j=1,2,\dotsc,d$, and any $k\geq 1$, define an integer

[TABLE]

If $\psi_{k,j}(x)>0$, there exists a unique word $Y_{k,j}(x)$ in $\{1,2,\dotsc,N\}^{\psi_{k,j}(x)}$ such that $x[1,k]\cap E\subset g_{j}^{-1}([Y_{k,j}(x)])$, which also means $g_{j}(x)\in[Y_{k,j}(x)]$ for $j=1,2,\dotsc,d$. We list two useful results as follows,

1. (P1)

Fix a positive integer $k$, if $x,y\in E$ with $x[1,k]=y[1,k]$, then $\psi_{k,j}(x)=\psi_{k,j}(y)$. Furthermore, if $\psi_{k,j}(x)=\psi_{k,j}(y)>0$, then $Y_{k,j}(x)=Y_{k,j}(y)$ for $j=1,2,\dotsc,d$. 2. (P2)

$\lim_{k\to\infty}\psi_{k,j}(x)=\infty$ for $j=1,2,\dotsc,d$. From the continuity of $g_{j}$ for $j=1,2,\dotsc,d$ and the fact that $\lim_{k\to\infty}\operatorname{diam}(x[1,k])=0$, it is clear that $\lim_{k\to\infty}\operatorname{diam}(g_{j}(x[1,k]\cap E))=0$. In other words, for any integer $M>0$, there exists $M_{1}>M$ such that $\operatorname{diam}(g_{j}(x[1,k]\cap E))<\frac{1}{N^{M}}$ for $j=1,2,\dotsc,d$ and any $k\geq M_{1}$. It implies that there exists a word $Y_{k,j}$ with length $M$ such that $x[1,k]\cap E\subset g_{j}^{-1}([Y_{k,j}])$ for $j=1,2,\dotsc,d$. So, for this $M$, $M_{1}$, and any $k\geq M_{1}$, it has that $\psi_{k,j}(x)\geq M$, which means $\lim_{k\to\infty}\psi_{k,j}(x)=\infty$ for $j=1,2,\dotsc,d$.

By (P2), there exists $m(x)>0$ such that $\psi_{k,j}(x)>0$ for $j=1,2,\dotsc,d$ and any $k\geq m(x)$. Therefore, for a sufficient large $k$ we can find the cylinder $Y_{k,j}(x)$ and in fact $\lim_{k\to\infty}\operatorname{diam}(Y_{k,j}(x))=0$ for $j=1,2,\dotsc,d$.

If $\psi_{k,j}(x)>0$ for any $j=1,2,\dotsc,d$, by (P1) we know that the set $\bigcup_{x\in E}\{Y_{k,j}(x)\colon 1\leq j\leq d\}$ is finite, since $\{x[1,k]\colon x\in E\}$ is finite. Thus, there exists $\mathbf{p}_{i}^{(k)}=(p_{i,1}^{(k)},p_{i,2}^{(k)},\dotsc,p_{i,l_{k}}^{(k)})\in\{1,2,\dotsc,{l_{k}}\}^{l_{k}}$ such that

[TABLE]

for any $x\in E$ with $\psi_{k,j}(x)>0$ for $j=1,2,\dotsc,d$. According to the construction of the map $\Delta_{N}$, there exists a positive integer $q_{k}=r^{(k)}_{i,1}$ such that

[TABLE]

hold for any $x\in E$. We claim that $\{q_{k}\}_{k=1}^{\infty}$ is the sequence that we want. As we have proved, for any $x\in D$ there exists $m(x)>0$ such that $\psi_{k,j}(x)>0$ holds for any $k\geq m(x)$, which means that both $\sigma^{j\cdot q_{k}}(x)$ and $g_{j}(x)$ are contained in the same cylinder $[Y_{k,j}(x)]$ for $j=1,2,\dotsc,d$. Then according to the fact that $\lim_{k\to\infty}\operatorname{diam}([Y_{k,j}(x)])=0$, we obtain $\lim_{k\to\infty}\sigma^{j\cdot q_{k}}(x)=g_{j}(x)$ for $j=1,2,\dotsc,d$. This ends the proof of Theorem 1.1.

Remark 3.2**.**

For the full shift over $N$ symbols, it is clear that $\rho(\sigma x,\sigma y)=N\rho(x,y)$ for any $x,y\in\Sigma_{N}$ with $\rho(x,y)<\frac{1}{N}$. Notice [FHYZ12, Lemma 5.1] and [FHYZ12, Lemma 5.4], it immediately turns out that the Hausdorff dimension of the multiply Xiong chaotic set $C$ is equal to its Bowen dimension entropy divided by the logarithm of $N$. Note that the Hausdorff dimension is dependent on the metric, but the Bowen dimension entropy is not.

4. Multiply Xiong chaotic set in the Gauss system

In this section we first study the Lebesgue measure of scrambled sets in the Gauss system and then provide a proof of Theorem  1.2 through the relation between the Gauss system and the full shift over countable symbols.

4.1. Proximal cells and Li-Yorke scrambled sets

Recall that the Gauss map $T\colon[0,1)\to[0,1)$ is defined by T(x)=\frac{1}{x}-\bigl{\lfloor}\frac{1}{x}\bigr{\rfloor} for $x\in(0,1)$ and $T(0)=0$ where $\lfloor\cdot\rfloor$ represents the integer part of a real number. The Gauss map induces an infinite continued fraction of every irrational number $x\in[0,1)$. Specifically, the continued fraction of $x\in[0,1)\setminus\mathbb{Q}$ is

[TABLE]

where a_{1}(x)=\bigl{\lfloor}\frac{1}{x}\bigr{\rfloor} and a_{n}(x)=\biggl{\lfloor}\frac{1}{T^{n-1}(x)}\biggr{\rfloor} for any $n\geq 2$. Although the Gauss map is not continuous, it still has some interesting dynamical properties discussed in the continuous dynamical system. And we are interested in the irrational part mostly. Hence, we call $([0,1)\setminus\mathbb{Q},T)$ the Gauss system and adopt the concepts in the continuous system to describe the dynamical properties for the Gauss system.

Let $\mathcal{B}$ be the Borel $\sigma$-algebra on $[0,1)$. We use $\mathcal{L}$ to denote the Lebesgue measure. It is well known that the Gauss map preserves the Gauss measure $\mu$ that is given by $\mu(A)=\frac{1}{\ln 2}\int_{A}\frac{1}{1+x}dx$ where the integration is with respect to the Lebesgue measure for any $A\in\mathcal{B}$. It is clear that the Gauss measure and the Lebesgue measure are equivalent, that is they have the same null sets and full measure sets. It is shown in [R61] that the Gauss map with the Gauss measure is exact.

Lemma 4.1**.**

The Gauss map sends Borel sets to Borel sets.

Proof.

Let $W$ be a Borel subset of $[0,1)$ and $W_{n}=W\cap[\frac{1}{n+1},\frac{1}{n})$ for any $n\geq 1$. Then $W$ can be written as $W=\bigcup_{n=1}^{\infty}W_{n}\cup(W\cap\{0\})$. Observe that $T|_{[\frac{1}{n+1},\frac{1}{n})}$ is a homeomorphism, $T(W)=\bigcup_{n=1}^{\infty}T(W_{n})\cup T(W\cap\{0\})$ is a Borel subset of $[0,1)$. ∎

Proposition 4.2**.**

For every irrational number $x$ in the Gauss system , $LY(x)=\{y\in[0,1)\setminus\mathbb{Q}\colon(x,y)$ $\textrm{ is scrambled}\}$ is residual and has full Lebesgue measure. In particular, every proximal cell is residual and has full Lebesgue measure.

Proof.

Now fix a point $x\in[0,1)\setminus\mathbb{Q}$, as the Gauss measure $\mu$ is exact, $\mu$ is strongly mixing by Proposition 2.2; In particular, $\mu\times\mu$ is ergodic. For any $k\in\mathbb{N}$, applying the well-known Birkhoff Ergodic theorem (see [WP]) to the indicator function $\chi_{[0,1/k]\times[1-1/k,1)}$ in the ergodic system $([0,1)\times[0,1),\mathcal{B}\times\mathcal{B},\mu\times\mu,T\times T)$, we obtain that for almost every $(z_{1},z_{2})\in[0,1)\times[0,1)$,

[TABLE]

It means that for a fixed positive integer $k$, there exists a measurable set $E_{k}\subset[0,1)\times[0,1)$ with $\mu\times\mu(E_{k})=1$ such that for any $(z_{1},z_{2})\in E_{k}$, there exists a positive integer $N_{k}=N_{k}(z_{1},z_{2})$ such that $\frac{1}{n}\sum_{i=0}^{n-1}\chi_{[0,1/k]\times[1-1/k,1)}(T^{i}\times T^{i}(z_{1},z_{2}))>0$ holds for any $n>N_{k}$. It implies that there exists a positive integer $0\leq l_{k}(z_{1},z_{2})\leq n-1$ such that $|T^{l_{k}}(z_{1})-T^{l_{k}}(z_{2})|\geq 1-\frac{2}{k}$. Then, $\bigcap_{k=1}^{\infty}E_{k}$ has full Lebesgue measure and is contained in $\{(z_{1},z_{2}):\limsup_{n\to\infty}|T^{n}z_{1}-T^{n}z_{2}|=1\}$. Set

[TABLE]

It is easy to see that the Borel set $A_{m}(x)$ is a $G_{\delta}$ subset of $[0,1)\setminus\mathbb{Q}$ for any integer $m\geq 2$. If $A_{m}(x)$ does not have full $\mu$-measure, then there exist $z_{1},z_{2}\in([0,1)\setminus A_{m}(x))\cap\bigcap_{k=1}^{\infty}E_{k}$ such that

[TABLE]

This is a contradiction, so $\mu(A_{m}(x))$ must be equal to $1$ for any $x\in[0,1)$ and $m\geq 2$.

Let $C(x)=\{y\in[0,1)\setminus\mathbb{Q}\colon\liminf_{n\to\infty}|T^{n}x-T^{n}y|=0\}$. It is easy to see that the Borel set $C(x)$ is a $G_{\delta}$ subset of $[0,1)\setminus\mathbb{Q}$. For $k\geq 1$, let

[TABLE]

and for $m\geq 1$, let

[TABLE]

It is clear that all $C_{k}$ and $C_{k,m}$ are also Borel sets. To prove $\mu(C(x))=1$, it is sufficient to show that $\mu(C_{k,m})=0$ for all $k,m\geq 1$ since $C_{k}=\bigcup_{m=1}^{\infty}C_{k,m}$ and $[0,1)\setminus C(x)=\bigcup_{k=1}^{\infty}C_{k}$. Assume by contradiction that there exist some $m,k\geq 1$ such that $\mu(C_{k,m})>0$. On one hand, for this fixed $k\geq 1$, it is easy to compute that the measure of the $\frac{1}{k}$-neighborhood of $T^{n}x$ is

[TABLE]

for any $n\geq 1$. On the other hand, by Lemma 4.1, $T^{n}C_{k,m}\in\mathcal{B}$ for any $n\geq 1$ and $\lim_{n\to\infty}\mu(T^{n}C_{k,m})=1$ from the exactness and Proposition 2.2. Set $\varepsilon=\frac{1}{\ln 2}\ln\frac{2k+1}{2k-1}$. There exists some $N>m$ such that $\mu(T^{n}C_{k,m})>1-\varepsilon$ for any $n\geq N$. Thus, we can pick a point $z$ in $T^{N}C_{k,m}\cap B(T^{N}x,\frac{1}{k})$. Write $N=m+l$ with $l\geq 1$. There exists $y\in T^{m}C_{k,m}$ such that $T^{l}y=z$, which implies that $|T^{l}y-T^{m+l}x|<\frac{1}{k}$. And there also exists $t\in C_{k,m}$ such that $y=T^{m}t$. According to the definition of $C_{k,m}$, it is easy to verify that

[TABLE]

which is a contradiction. Therefore, $\mu(C_{k,m})$ must be zero and $\mu(C(x))=1$. Note that $LY(x)$ is exactly the set $\bigcap_{m=1}^{\infty}A_{m}(x)\cap C(x)$, which has full Lebesgue measure and contains a dense $G_{\delta}$ subset. This ends the proof. ∎

Recall that a dynamical system $(X,T)$ is Li-Yorke sensitive if there exists a sensitive constant $\delta>0$ such that for any $x\in X$ and any $\varepsilon>0$, there exists $y$ in the $\varepsilon$-neighborhood $B(x,\varepsilon)$ such that $(x,y)$ is proximal and $\lim\sup_{n\to\infty}d(T^{n}x,T^{n}y)\geq\delta$. A scrambled set $S$ is call maximal if $S$ is maximal in the inclusion relation among all scrambled sets. The following two corollaries are clear.

Corollary 4.3**.**

The Gauss system is Li-Yorke sensitive.

Corollary 4.4**.**

Every maximal scrambled set in the Gauss system is uncountable.

Proposition 4.5**.**

Every measurable scrambled set in the Gauss system has Lebesgue measure zero.

Proof.

Assume that there exists a scrambled set $B\subset[0,1)\setminus\mathbb{Q}$ with positive Lebesgue measure, it is clear that $B$ also has positive Gauss measure. Choose a Lebesgue measurable subset $B_{1}$ of $B$ such that $\mu(B_{1})>0$ and $\mu(B\setminus B_{1})>0$. By Lemma 4.1 and the exactness of the Gauss map, we have $\lim_{n\to\infty}\mu(T^{n}(B_{1}))=1$ and $\lim_{n\to\infty}\mu(T^{n}(B\setminus B_{1}))=1$ for $n\in\mathbb{N}$. This means that there exists $N\in\mathbb{N}$ such that $\mu(T^{N}(B_{1}))>\frac{1}{2}$ and $\mu(T^{N}(B\setminus B_{1}))>\frac{1}{2}$. It implies $T^{N}(B_{1})\cap T^{N}(B\setminus B_{1})\neq\emptyset$. This is a contradiction because $B$ is scrambled and $T^{N}|_{B}$ must be injective. ∎

Note that a multiply Xiong chaotic set is scrambled, we have the following corollary.

Corollary 4.6**.**

Every measurable multiply Xiong chaotic set in the Gauss system has Lebesgue measure zero.

4.2. multiply Xiong chaotic set in the

full shift over countable symbols

Let $\Sigma_{\infty}=\mathbb{N}^{\infty}$. There exists a natural bijection $\phi\colon\Sigma_{\infty}\to[0,1)\setminus\mathbb{Q}$ by $\phi(a_{1}a_{2}\dotsc)=[a_{1},a_{2},\dotsc]$. Endow $\mathbb{N}$ with the discrete topology. Then the product space $\Sigma_{\infty}$ is metrizable, separable, not compact but complete. A compatible metric $\rho$ on $\Sigma_{\infty}$ can be defined as follow: for any $x=x_{1}x_{2}\dotsb$ and $y=y_{1}y_{2}\dotsb\in\Sigma_{\infty}$,

[TABLE]

It is not hard to show that $\phi$ is a homeomorphism between $(\Sigma_{\infty},\sigma)$ and $([0,1)\setminus\mathbb{Q},T)$ with $\phi\circ\sigma=T\circ\phi$. But it should be noticed that neither $\phi$ nor $\phi^{-1}$ is Lipschitz continuous. The shift map on $\Sigma_{\infty}$ is defined by $\sigma\colon\Sigma_{\infty}\to\Sigma_{\infty}$, $\sigma(x)=x_{2}x_{3}\dotsb$ for any $x=x_{1}x_{2}\dotsb$. It is clear that $\sigma$ is continuous. The pair $(\Sigma_{\infty},\sigma)$ is called the full shift over countable symbols. The definitions of word, prefix and cylinder etc. are similar to the definitions in full shift over finite symbols.

Proposition 4.7**.**

If $E\subset\Sigma_{\infty}$ is a multiply Xiong chaotic set in the full shift over countable symbols $(\Sigma_{\infty},\sigma)$, then $\phi(E)$ is a multiply Xiong chaotic set in the Gauss system. Moreover, for any $x\in\Sigma_{\infty}$, if $E$ is a subset of $MProx(x,\sigma)$, then $\phi(E)$ is contained in $MProx(\phi(x),T)$.

Proof.

For any $d\in\mathbb{N}$, any non-empty subset $M\subset\phi(E)$, and any continuous maps $g_{1},g_{2},\dotsc,$ $g_{d}:M\to[0,1)$, define continuous maps $F_{j}=\phi^{-1}\circ g_{j}\circ\phi:\phi^{-1}(M)\to\Sigma_{\infty}$ for $j=1,2,\dotsc,d$. Since $E$ is a multiply Xiong chaotic set in the full shift over countable symbols, for the non-empty subset $\phi^{-1}(M)\subset E$ and every continuous map $F_{j}:\phi^{-1}(M)\to\Sigma_{\infty}$ , there exists an increasing sequence $\{r_{n}\}$ such that

[TABLE]

It implies

[TABLE]

for any $x\in M$ and $j=1,2,\dotsc,d$. The second result is clear. ∎

By Proposition 4.7, in order to study the multiply Xiong chaotic set in the Gauss system, we need to construct a multiply Xiong chaotic set in the full shift over countable symbols. As the idea is similar to the construction in the proof of Theorem 1.1, we only sketch the construction and the details are left to interested readers.

Lemma 4.8**.**

[WT07, Lemma 2]* Let $W=\prod_{i=1}^{\infty}\{1,2,\dotsc,i\}$ be a compact subset of $\Sigma_{\infty}$. We have $\operatorname{dim_{H}}(W\cap U)=\infty$ where $U$ is a cylinder of $\Sigma_{\infty}$ with $W\cap U$ not empty.*

Lemma 4.9**.**

[WT07, Lemma 3]* Let $A=\{a_{n}\}_{n=1}^{\infty}$ be a sequence of strictly increasing positive numbers. Define a map $\Gamma_{A}:\Sigma_{\infty}\to\Sigma_{\infty}$, $\Gamma_{A}(x)=x_{1}x_{2}\dotsb x_{a_{1}-1}x_{a_{1}+1}\dotsb x_{a_{n}-1}x_{a_{n}+1}\dotsb$. Let $Y$ be a subset of $\Sigma_{\infty}$. If the density of $A$ is less than $1$ and $\operatorname{dim_{H}}(\Gamma_{A}(Y))=\infty$, then $\operatorname{dim_{H}}(Y)=\infty$.*

Proposition 4.10**.**

In the full shift $(\Sigma_{\infty},\sigma)$ over countable symbols, for every $z\in\Sigma_{\infty}$, the multiply proximal cell of $z$ contains a multiply Xiong chaotic sets with full Hausdorff dimension everywhere.

Proof.

Let $n\in\mathbb{N}$ and $l_{n}=(n^{n})^{n^{n}}$. We list all the self-maps on $\{1,2,\dotsc,n\}^{n}$ as $\varphi^{(n)}_{1},\varphi^{(n)}_{2},\dotsc,\varphi^{(n)}_{l_{n}}$. We can define a map $\Delta_{\infty}:\Sigma_{\infty}\to\Sigma_{\infty}$ similar to $\Delta_{N}$ in the proof of Theorem 1.1. Set $W=\prod_{i=1}^{\infty}\{1,2,\dots,i\}$. For any $n\geq 1$ and $v=v_{1}v_{2}\dotsb v_{n}\in\mathbb{N}^{n}$, we can also define a map $\theta_{v}:\Sigma_{\infty}\to\Sigma_{\infty}$ by $\theta_{v}(x)=v_{1}v_{2}\dotsb v_{n}x_{n+1}\dotsb$ for any $x\in\Sigma_{\infty}$. Set $\{w_{n}:n\geq 1\}=\bigcup_{k=1}^{\infty}\mathbb{N}^{k}$, $B_{n}=W\cap[12\dotsc n1]$, $B=\bigcup_{n=1}^{\infty}\Delta_{\infty}(B_{n})$ and $C=\bigcup_{n=1}^{\infty}\theta_{w_{n}}\circ\Delta_{\infty}(B_{n})$. It is easy to see that each $B_{n}$ is not empty and pairwise disjoint closed subset, so is $\Delta_{\infty}(B_{n})$ for any $n\geq 1$.

Similarly, we can also define a continuous map $\xi\colon B\to C$ such that $\xi|_{\Delta_{\infty}(B_{n})}=\theta_{w_{n}}|_{\Delta_{\infty}(B_{n})}$. Obviously, $C$ is contained in the multiply proximal cell of $z$. We proceed to show that $C$ is with full Hausdorff dimension everywhere and then show its chaotic property. For any non-empty open subset $U$ of $\Sigma_{\infty}$, there exists some $w_{n}$ such that $[w_{n}]\subset U$, so $C\cap U$ contains $\theta_{w_{n}}\circ\Delta_{\infty}(B_{n})$. According to Lemma 4.8, Lemma 4.9, and the same explanation in the proof of Theorem 1.1, we know that $\operatorname{dim_{H}}(\theta_{w_{n}}\circ\Delta_{\infty}(B_{n}))=\infty$.

The remaining part is to show that $C$ is a multiply Xiong chaotic set. Most of this part is similar to the full shift over finite symbols. We only explain the difference here. As $\xi\colon B\to C$ is continuous, it is sufficient to show that $B$ is a multiply Xiong chaotic set. Let $E\subset B$ and $g_{j}\colon E\to\Sigma_{\infty}$ be a continuous map for $j=1,2,\dotsc,d$. Similarly, for $j=1,2,\dotsc,d$, any $x\in E$, and any $k\geq 1$, we can define an integer $\psi_{k,j}(x)$ as the proof of Theorem 1.1 and all the results about $\psi_{k,j}(x)$ are also valid. Observe that $B$ is a subset of the compact set $\Delta_{\infty}(W)$, the set $\{x[1,k]\colon x\in E\}$ is finite. By (P1), we know that the set $\bigcup_{x\in E}\{Y_{k,j}(x)\colon 1\leq j\leq d\}$ is finite. Then we can carry on applying the method in the proof of Theorem 1.1 to show that $C$ is a multiply Xiong chaotic set in $(\Sigma_{\infty},\sigma)$.

∎

4.3. Proof of Theorem 1.2

By Propositions 4.7 and 4.10, we should estimate the Hausdorff dimension of the image of the multiply Xiong chaotic under the map $\phi$. To this end, we need some results of the continued fraction and we refer the reader to [Io2002] for more details. Assume that the infinite continued fraction of $x\in[0,1)\setminus\mathbb{Q}$ is $x=[a_{1},a_{2},\dotsc]$. For any $n\in\mathbb{N}$, we call the rational number $[a_{1},a_{2},\dotsc,a_{n}]$ the $n$-th convergent of $x$ and denote it by

[TABLE]

where $p_{n},q_{n}\in\mathbb{N}$ and $(p_{n},q_{n})=1$. If we set $p_{0}=q_{-1}=0$, $p_{-1}=q_{0}=1$, then for any $n\in\mathbb{N}$,

[TABLE]

For any $n\in\mathbb{N}$ and $i_{1}i_{2}\dotsc i_{n}\in\mathbb{N}^{n}$, the fundamental interval is defined by

[TABLE]

The endpoints of $I^{n}(i_{1},i_{2},\dotsc,i_{n})$ are $\frac{p_{n}+p_{n-1}}{q_{n}+q_{n-1}}$ and $\frac{p_{n}}{q_{n}}$ where $p_{n}=p_{n}(i_{1},i_{2},\dotsc,i_{n})$ and $q_{n}=q_{n}(i_{1},i_{2},\dotsc,i_{n})$. More precisely, in the case when $n$ is odd,

[TABLE]

and in the case when $n$ is even,

[TABLE]

Whatever, the Lebesgue measure of the interval $I^{n}(i_{1},i_{2},\dotsc,i_{n})$ is $\frac{1}{q_{n}(q_{n}+q_{n-1})}$. We will need the following useful results.

Lemma 4.11** ([Wu2006]).**

For any $n\geq 1$ and $1\leq k\leq n$, it has that

[TABLE]

Lemma 4.12** ([Hu2014]).**

There exists a positive number $\lambda$ such that

[TABLE]

where $\mu=u_{1}u_{2}\dotsc u_{n}\in\mathbb{N}^{n}$ and $\nu=v_{1}v_{2}\dotsc v_{k}\in\mathbb{N}^{k}$ for any $k,n\in\mathbb{N}$.

Theorem 4.13** ([J28]).**

For any $k>8$,

[TABLE]

Inspired by the proof of Theorem 4.13 in [J28], we have the following result.

Proposition 4.14**.**

Let $W=\prod_{i=1}^{\infty}\{1,2,\dotsc,i\}$. For any $l\geq 1$, any $i_{1},i_{2},\dotsc,i_{l}\in\mathbb{N}$, and any integer $k>\max\{4,i_{1},i_{2},\dotsc,i_{l},l+1\}$,

[TABLE]

Proof.

For any $n\geq 1$, denote

[TABLE]

For any $n\geq k$, denote

[TABLE]

and

[TABLE]

The following properties are obvious by the definitions:

1. (1)

$I^{n+1}_{i_{1},i_{2},\dotsc,i_{l},a_{l+1},\dotsc,a_{n+1}}\subset I^{n}_{i_{1},i_{2},\dotsc,i_{l},a_{l+1},\dotsc,a_{n}}$ for any $n\geq k$. 2. (2)

$M^{n}_{i_{1},i_{2},\dotsc,i_{l},a_{l+1},\dotsc,a_{n}}\subset I^{n}_{i_{1},i_{2},\dotsc,i_{l},a_{l+1},\dotsc,a_{n}}$ for any $n\geq k$. 3. (3)

If $a_{n+1}\leq k$, then $I^{n+1}_{i_{1},i_{2},\dotsc,i_{l},a_{l+1},\dotsc,a_{n+1}}\subset M^{n}_{i_{1},i_{2},\dotsc,i_{l},a_{l+1},\dotsc,a_{n}}$ for any $n\geq k$. 4. (4)

$W_{n}=V_{n+1}$ for any $n\geq k$. 5. (5)

$V_{k}\supset W_{k}=V_{k+1}\supset W_{k+1}=V_{k+2}\supset\dotsb$

It is clear that

[TABLE]

which is closed and perfect because $V_{n}$ is not a singular and $\lim_{n\to\infty}\operatorname{diam}(V_{n})=0$. It is clear that $\phi\circ\theta_{i_{1}i_{2}\dotsc i_{l}}(W)\cap\phi(\Sigma_{k})$ is compact. Let $\delta>0$ and $L$ be a positive integer. Assume that $\mathcal{G}=\{B_{1},B_{2},\dotsc,B_{L}\}$ is a finite $\delta$-cover (the diameter of each element of $\mathcal{G}$ is less than $\delta$) of $\phi\circ\theta_{i_{1}i_{2}\dotsc i_{l}}(W)\cap\phi(\Sigma_{k})$ with $B_{j}\cap\phi\circ\theta_{i_{1}i_{2}\dotsc i_{l}}(W)\cap\phi(\Sigma_{k})$ containing infinite points for $j=1,2,\dotsc,L$. In the closed set $\phi\circ\theta_{i_{1}i_{2}\dotsc i_{l}}(W)\cap\phi(\Sigma_{k})$ we can pick $b_{j}=\inf(B_{j}\cap\phi\circ\theta_{i_{1}i_{2}\dotsc i_{l}}(W)\cap\phi(\Sigma_{k}))$ and $c_{j}=\sup(B_{j}\cap\phi\circ\theta_{i_{1}i_{2}\dotsc i_{l}}(W)\cap\phi(\Sigma_{k}))$ for $j=1,2,\dotsc,L$. Thus, $\mathcal{G}^{\prime}=\{[b_{j},c_{j}]:j=1,2,\dotsc,L\}$ is still a $\delta$-cover of $\phi\circ\theta_{i_{1}i_{2}\dotsc i_{l}}(W)\cap\phi(\Sigma_{k})$ by replacing $B_{i}$ with $[b_{i},c_{i}]$ in $\mathcal{G}$ for $j=1,2,\dotsc,L$. It is clear that

[TABLE]

Since both $b_{j}$ and $c_{j}$ are elements of

[TABLE]

we can set

[TABLE]

It is not hard to show that the following result: For every $[b_{j},c_{j}]\in\mathcal{G}^{\prime}$ with $j=1,2,\dotsc,L$, there exist $m\geq k$, $a_{l+1},a_{l+2},\dotsc,a_{m}$ with $1\leq a_{q}\leq q$ for $l+1\leq q\leq k-1$, and $a_{q}\leq k$ for any $k\leq q\leq m$ such that $[b_{j},c_{j}]\subset M^{m}_{i_{1},i_{2},\dotsc.i_{l},a_{l+1},\dotsc,a_{m}}$. And there also exist two distinct integers $s$ and $t$ with $1\leq s\neq t\leq k$ such that $[b_{j},c_{j}]\cap M^{m+1}_{i_{1},i_{2},\dotsc.i_{l},a_{l+1},,a_{m},s}\neq\emptyset$ and $[b_{j},c_{j}]\cap M^{m+1}_{i_{1},i_{2},\dotsc.i_{l},a_{l+1},\dotsc,a_{m},t}\neq\emptyset$.

In the following, as only the endpoints of a close interval are needed, we use $(a,b)$ to denote $[a,b)$ and $[b,a)$ for convenience. Observe that for any $n\geq k$,

[TABLE]

where $p_{n-1}=p_{n-1}(i_{1},i_{2},\dotsc,i_{l},a_{l+1},\dotsc,a_{n-1})$, $q_{n-1}=q_{n-1}(i_{1},i_{2},\dotsc,i_{l},a_{l+1},\dotsc,a_{n-1})$, $p_{n}=p_{n}(i_{1},i_{2},\dotsc,i_{l},a_{l+1},\dotsc,a_{n})$, and $q_{n}=q_{n}(i_{1},i_{2},\dotsc,i_{l},a_{l+1},\dotsc,a_{n})$. As a result, the distance between $M^{m+1}_{i_{1},i_{2},\dotsc,i_{l},a_{l+1},\dotsc,a_{m},s}$ and $M^{m+1}_{i_{1},i_{2},\dotsc,i_{l},a_{l+1},\dotsc,a_{m},t}$ is not less than the minimum of

[TABLE]

and

[TABLE]

Furthermore, both (4.1) and (4.2) are not less than $\frac{1}{3k^{3}q_{m}(q_{m}+q_{m-1})}$ , because

[TABLE]

and similarly

[TABLE]

Therefore, it can be concluded that

[TABLE]

from the fact that

[TABLE]

So far, it has been proved that for any $[b_{j},c_{j}]\in\mathcal{G}^{\prime}$, there exists an interval $I^{m}(i_{1},i_{2},\dotsc i_{l},$ $\dotsc,a_{l+1},\dotsc,a_{m})$ corresponding to it where $m$ and $a_{l+1},a_{l+2},\dotsc,a_{m}$ are all related to $[b_{j},c_{j}]$. Next, replacing $[b_{j},c_{j}]$ by the interval $I^{m}(i_{1},i_{2},\dotsc i_{l},\dotsc,a_{l+1},\dotsc,a_{m})$, we get another family $\mathcal{C}$ that is a $3k^{3}\delta$-cover of $\phi\circ\theta_{i_{1}i_{2}\dotsc i_{l}}(W)\cap\phi(\Sigma_{k})$.

Hitherto, we have known that for any $p>0$ and any finite $\delta$-cover $\mathcal{G}$ of $\phi\circ\theta_{i_{1}i_{2}\dotsc i_{l}}(W)\cap\phi(\Sigma_{k})$, there exists a $3k^{3}\delta$-cover $\mathcal{C}$ of $\phi\circ\theta_{i_{1}i_{2}\dotsc i_{l}}(W)\cap\phi(\Sigma_{k})$ satisfying

[TABLE]

Moreover, set

[TABLE]

Obviously, $h\geq k$. If there exist two intervals, for example some $I^{r},I^{n}\in\mathcal{C}$ with $r>n$ and $I^{r}\subset I^{n}$, then remove $I^{r}$ from $\mathcal{C}$. We can get another $3k^{3}\delta$-cover $\mathcal{M}$ by such a further repetition. It is clear that any two distinct intervals $I^{r}$ and $I^{n}$ in $\mathcal{M}$ do not intersect. Set

[TABLE]

Then $p\geq h\geq k$. According to the definition, there exist $a_{l+1},a_{l+2},\dotsc,a_{p}$ with $a_{i}\leq i$ for $l+1\leq i\leq k-1$ and $a_{k},a_{k+1},\dotsc,a_{p}\leq k$ such that $I^{p}(i_{1},i_{2},\dotsc,i_{l},a_{l+1},\dotsc,a_{p})\in\mathcal{M}$, which implies that $I^{p}(i_{1},i_{2},\dotsc,i_{l},a_{l+1},\dotsc,1)$, $I^{p}(i_{1},i_{2},\dotsc,i_{l},a_{l+1},\dotsc,2)$, $\dotsc$, $I^{p}(i_{1},i_{2},\dotsc,i_{l},a_{l+1},$ $\dotsc,k)$ are certainly all in $\mathcal{M}$.

**Claim. ** Let $s=1-\frac{4}{k\ln 2}$. For any $n\geq h+1$, any $a_{l+1},a_{l+2},\dotsc,a_{p}$ with $a_{i}\leq i$ for $l+1\leq i\leq k-1$, and any $a_{k},a_{k+1},\dotsc,a_{p}\leq k$, it has that

[TABLE]

**Proof of the claim. ** At first, compute

[TABLE]

So,

[TABLE]

where $\beta=\dfrac{kq_{n-1}+kq_{n-2}}{kq_{n-1}+q_{n-1}+q_{n-2}}$ and $\dfrac{1}{2}<\beta<2$. If $s=1-\dfrac{4}{k\ln 2}$, then

[TABLE]

We obtain that \biggl{(}1-\frac{\beta}{k}\biggr{)}\times 2^{1-s}\geq 1 for any $\frac{1}{2}<\beta<2$. So, it is allowed to replace the following $k$ intervals :

[TABLE]

by one interval $I^{k-1}(i_{1},i_{2},\dotsc,i_{l},a_{l+1},\dotsc,a_{k-1})$. Do such replacement until we get a $3k^{3}\delta$-cover $\mathcal{J}$ in which the longest interval is some $I^{h}$. Through the process, it guarantees that

[TABLE]

Thus, for any positive number $p<\delta$ and any $p$-cover $\mathcal{B}$, it has that

[TABLE]

which means $\operatorname{dim_{H}}(\phi\circ\theta_{i_{1}i_{2}\dotsc i_{l}}(W)\cap\phi(\Sigma_{k}))\geq s=1-\frac{4}{k\ln 2}$. ∎

Now we are ready to prove Theorem 1.2.

Proof of Theorem 1.2.

Let $W=\prod_{i=1}^{\infty}\{1,2,\dotsc,i\}$. According to the proof of Proposition 4.10, we can construct a multiply Xiong chaotic set $\bigcup_{n=1}^{\infty}\theta_{w_{n}}\circ\Delta_{\infty}(W)$ contained in the multiply proximal cell of $\phi^{-1}(z)$. By Proposition 4.7, let $L=\phi(C)=\bigcup_{n=1}^{\infty}\phi(\theta_{w_{n}}\circ\Delta_{\infty}(W))$ be a multiply Xiong chaotic set contained in $MProx(z,T)$ in the Gauss system. Now, it remains to show that $L\subset[0,1)$ is a subset with full Hausdorff dimension everywhere. Note that $\phi\circ\theta_{i_{1}i_{2}\dotsb i_{n}}\circ\Delta_{\infty}(W)$ is exactly the set $L\cap I^{n}(i_{1},i_{2},\dotsc,i_{n})$ for any $n\geq 1$ and any $i_{1},i_{2},\dotsb,i_{n}\in\mathbb{N}$, it is sufficient to show that $\operatorname{dim_{H}}(\phi\circ\theta_{i_{1}i_{2}\dotsb i_{n}}\circ\Delta_{\infty}(W))=1$. We divide the remaining part of the proof into three parts. First, define a map

[TABLE]

It is not hard to see that $f=\phi\circ\theta_{i_{1}i_{2}\dotsc i_{n}}\circ\Delta_{\infty}^{-1}\circ\theta_{i_{1}i_{2}\dotsb i_{n}}^{-1}\circ\phi^{-1}$ and $f$ is bijective and continuous.

Second, recall the subset $D$ of $\mathbb{N}$ with zero density in the construction of the map $\Delta_{\infty}$ in the proof of Theorem 1.1. Let $\tau(n)=\#(\{a_{i}\leq n:a_{i}\in D\})$. It can be inferred that $\lim\limits_{n\to\infty}\frac{1+c\cdot\tau(n)}{n-\tau(n)-1}=0$ where $c$ is a non-negative constant. For a fixed real number $k>0$, if we choose $c=\frac{2\ln(k+1)}{\ln 2}$, then for any $\varepsilon>0$, there exists a large enough $N\in\mathbb{N}$ such that for any $n\geq N$,

[TABLE]

which means

[TABLE]

Third, we are going to prove that for $\varepsilon>0$, $f$ satisfies the locally $\frac{1}{1+\varepsilon}$-Hölder condition. Since $\Delta_{\infty}(W)$ is compact, the set $\mathcal{D}_{i}=\{x[i]:x\in\Delta_{\infty}(W)\}$ is finite for any $i\geq 1$. Set

[TABLE]

For a pair of two distinct points $\beta=[i_{1},i_{2},\dotsb,i_{n},b_{n+1},\dotsb]$ and $\gamma=[i_{1},i_{2},\dotsb,i_{n},c_{n+1},\dotsc]$ in $\phi\circ\theta_{i_{1}i_{2}\dotsc i_{n}}\circ\Delta_{\infty}(W)$ with $|\beta-\gamma|<r$, there exists some $m\geq N$ such that $b_{i}=c_{i}$ for any $1\leq i\leq m$ but $b_{m+1}\neq c_{m+1}$ with $N+1\leq m+1\not\in D$. Obviously, both $b_{m+1}$ and $c_{m+1}$ are not more than some $k$ since $\phi\circ\theta_{i_{1}i_{2}\dots i_{n}}\circ\Delta_{\infty}(W)$ is compact. If $m$ is odd, $\gamma$ is less than $\beta$ and the interval $I^{m+2}(b_{1},b_{2},\dotsc,b_{m+1},k+1)$ is contained in $[\gamma,\beta]$. Otherwise, $\gamma$ is larger than $\beta$ and $I^{m+2}(b_{1},b_{2},\dotsc,b_{m+1},k+1)$ is contained in $[\beta,\gamma]$. So, by Lemma 4.12 ,

[TABLE]

Set $\eta$ be the word by erasing the items with position in $D$ from $i_{1},\dotsb,i_{n},b_{n+1},\dotsb,b_{m}$. Let $M_{i_{1},i_{2},\dotsc,i_{n},\dotsc,b_{m}}=\max\{i_{1},i_{2},\dotsc,i_{n},\dotsc,b_{m}\}+1$. Note that $m\geq N$, by Lemma 4.11,

[TABLE]

Combine with (4.3), it is clear that

[TABLE]

It can be concluded from Lemma 2.1 that

[TABLE]

for any $i_{1},i_{2},\dotsc,i_{n}\in\mathbb{N}$. Hence,

[TABLE]

for any $i_{1},i_{2},\dotsc,i_{n}\in\mathbb{N}$. By Proposition 4.14,

[TABLE]

We obtain

[TABLE]

for any $k>\max\{i_{1},i_{2},\dotsc,i_{n},n+1\}$. Passing $k\to\infty$, this implies

[TABLE]

for any $i_{1},i_{2},\dotsc,i_{n}\in\mathbb{N}$. ∎

Acknowledgments

The work was supported by NNSF of China (Nos. 11471125, 11871228 and 11771264) and NSF of Guangdong Province(2018B030306024).

{bibsection}

• [\resetbiblist99] AkinE.GlasnerE.HuangW.ShaoS.YeX.Sufficient conditions under which a transitive system is chaoticErgodic Theory Dynam. Systems30201051277–1310ISSN 0143-3857Review MathReviews@article{MR2718894, author = {Akin, E.}, author = {Glasner, E.}, author = {Huang, W.}, author = {Shao, S.}, author = {Ye, X.}, title = {Sufficient conditions under which a transitive system is chaotic}, journal = {Ergodic Theory Dynam. Systems}, volume = {30}, date = {2010}, number = {5}, pages = {1277–1310}, issn = {0143-3857}, review = {\MR{2718894}}} AuslanderJosephOn the proximal relation in topological dynamicsProc. Amer. Math. Soc.111960890–895ISSN 0002-9939Review MathReviews@article{Au60, author = {Auslander, Joseph}, title = {On the proximal relation in topological dynamics}, journal = {Proc. Amer. Math. Soc.}, volume = {11}, date = {1960}, pages = {890–895}, issn = {0002-9939}, review = {\MR{0164335}}} AkinEthanKolyadaSergiĭLi-yorke sensitivityNonlinearity16200341421–1433ISSN 0951-7715Review MathReviews@article{AK03, author = {Akin, Ethan}, author = {Kolyada, Sergi\u\i}, title = {Li-Yorke sensitivity}, journal = {Nonlinearity}, volume = {16}, date = {2003}, number = {4}, pages = {1421–1433}, issn = {0951-7715}, review = {\MR{1986303}}} BalibreaFranciscoJiménez LópezVíctorThe measure of scrambled sets: a surveyActa Univ. M. Belii Ser. Math.719993–11ISSN 1338-712XReview MathReviews@article{BL99, author = {Balibrea, Francisco}, author = {Jim'{e}nez L'{o}pez, V'{i}ctor}, title = {The measure of scrambled sets: a survey}, journal = {Acta Univ. M. Belii Ser. Math.}, number = {7}, date = {1999}, pages = {3–11}, issn = {1338-712X}, review = {\MR{1766951}}} BlanchardFranc̣oisHuangWenEntropy sets, weakly mixing sets and entropy capacityDiscrete Contin. Dyn. Syst.2020082275–311ISSN 1078-0947Review MathReviews@article{BH08, author = {Blanchard, Fran\d{c}ois}, author = {Huang, Wen}, title = {Entropy sets, weakly mixing sets and entropy capacity}, journal = {Discrete Contin. Dyn. Syst.}, volume = {20}, date = {2008}, number = {2}, pages = {275–311}, issn = {1078-0947}, review = {\MR{2358261}}} BlanchardFranc̣oisHuangWenSnohaL’ubomírTopological size of scrambled setsColloq. Math.11020082293–361ISSN 0010-1354Review MathReviews@article{BHS08, author = {Blanchard, Fran\d{c}ois}, author = {Huang, Wen}, author = {Snoha, L'ubom'{i}r}, title = {Topological size of scrambled sets}, journal = {Colloq. Math.}, volume = {110}, date = {2008}, number = {2}, pages = {293–361}, issn = {0010-1354}, review = {\MR{2353910}}} BruinHenkJiménez LópezVíctorOn the lebesgue measure of li-yorke pairs for interval mapsComm. Math. Phys.29920102523–560ISSN 0010-3616Review MathReviews@article{B09, author = {Bruin, Henk}, author = {Jim'enez L'opez, V'\i ctor}, title = {On the Lebesgue measure of Li-Yorke pairs for interval maps}, journal = {Comm. Math. Phys.}, volume = {299}, date = {2010}, number = {2}, pages = {523–560}, issn = {0010-3616}, review = {\MR{2679820}}} FangChunHuangWenYiYingfeiZhangPengfeiDimensions of stable sets and scrambled sets in positive finite entropy systemsErgodic Theory Dynam. Systems3220122599–628ISSN 0143-3857Review MathReviews@article{FHYZ12, author = {Fang, Chun}, author = {Huang, Wen}, author = {Yi, Yingfei}, author = {Zhang, Pengfei}, title = {Dimensions of stable sets and scrambled sets in positive finite entropy systems}, journal = {Ergodic Theory Dynam. Systems}, volume = {32}, date = {2012}, number = {2}, pages = {599–628}, issn = {0143-3857}, review = {\MR{2901362}}} FurstenbergH.Recurrence in ergodic theory and combinatorial number theoryM. B. Porter LecturesPrinceton University Press, Princeton, N.J.1981xi+203ISBN 0-691-08269-3Review MathReviews@book{Fu81, author = {Furstenberg, H.}, title = {Recurrence in ergodic theory and combinatorial number theory}, note = {M. B. Porter Lectures}, publisher = {Princeton University Press, Princeton, N.J.}, date = {1981}, pages = {xi+203}, isbn = {0-691-08269-3}, review = {\MR{603625}}} FalconerKennethFractal geometry3Mathematical foundations and applicationsJohn Wiley & Sons, Ltd., Chichester2014xxx+368ISBN 978-1-119-94239-9Review MathReviews@book{Fa2003, author = {Falconer, Kenneth}, title = {Fractal geometry}, edition = {3}, note = {Mathematical foundations and applications}, publisher = {John Wiley & Sons, Ltd., Chichester}, date = {2014}, pages = {xxx+368}, isbn = {978-1-119-94239-9}, review = {\MR{3236784}}} HuHuiYuYueliOn schmidt’s game and the set of points with non-dense orbits under a class of expanding mapsJ. Math. Anal. Appl.41820142906–920ISSN 0022-247XReview MathReviews@article{Hu2014, author = {Hu, Hui}, author = {Yu, Yueli}, title = {On Schmidt's game and the set of points with non-dense orbits under a class of expanding maps}, journal = {J. Math. Anal. Appl.}, volume = {418}, date = {2014}, number = {2}, pages = {906–920}, issn = {0022-247X}, review = {\MR{3206688}}} HuangWenLiJianYeXiangdongZhouXiaoyaoPositive topological entropy and $\Delta$-weakly mixing setsAdv. Math.3062017653–683ISSN 0001-8708Review MathReviews@article{HLYZ17, author = {Huang, Wen}, author = {Li, Jian}, author = {Ye, Xiangdong}, author = {Zhou, Xiaoyao}, title = {Positive topological entropy and $\Delta$-weakly mixing sets}, journal = {Adv. Math.}, volume = {306}, date = {2017}, pages = {653–683}, issn = {0001-8708}, review = {\MR{3581313}}} HuangWenShaoSongYeXiangdongMixing and proximal cells along sequencesNonlinearity17200441245–1260ISSN 0951-7715Review MathReviews@article{HW04, author = {Huang, Wen}, author = {Shao, Song}, author = {Ye, Xiangdong}, title = {Mixing and proximal cells along sequences}, journal = {Nonlinearity}, volume = {17}, date = {2004}, number = {4}, pages = {1245–1260}, issn = {0951-7715}, review = {\MR{2069703}}} IosifescuMariusKraaikampCorMetrical theory of continued fractionsMathematics and its Applications547Kluwer Academic Publishers, Dordrecht2002xx+383ISBN 1-4020-0892-9Review MathReviews@book{Io2002, author = {Iosifescu, Marius}, author = {Kraaikamp, Cor}, title = {Metrical theory of continued fractions}, series = {Mathematics and its Applications}, volume = {547}, publisher = {Kluwer Academic Publishers, Dordrecht}, date = {2002}, pages = {xx+383}, isbn = {1-4020-0892-9}, review = {\MR{1960327}}} JarníkVojtěchZur metrischen theorie der diophantischen approximationenGermanPrace Matematyczno-Fizyczne361928Linkpol1191–106ISSN 0025-5874Review MathReviews@article{J28, author = {{Jarn' ik}, {Vojt\v ech}}, title = {Zur metrischen Theorie der diophantischen Approximationen}, language = {German}, journal = {Prace Matematyczno-Fizyczne}, volume = {36}, date = {1928}, url = {http://eudml.org/doc/215473}, language = {pol}, issue = {1}, number = {1}, pages = {91–106}, issn = {0025-5874}, review = {\MR{1544787}}} LiuKairan$\Delta$-Weakly mixing subset in positive entropy actions of a nilpotent groupJ. Differential Equations2019ISSN 0022–0396DocumentLink@article{Liu2019, author = {Liu, Kairan}, title = {$\Delta$-weakly mixing subset in positive entropy actions of a nilpotent group}, journal = {J. Differential Equations}, date = {2019}, issn = {0022–0396}, doi = {doi.org/10.1016/j.jde.2019.01.018}, url = {www.sciencedirect.com/science/article/pii/S0022039619300403}} LiuWeibinLiBingChaotic and topological properties of continued fractionsJ. Number Theory1742017369–383ISSN 0022-314XReview MathReviews@article{Li2017, author = {Liu, Weibin}, author = {Li, Bing}, title = {Chaotic and topological properties of continued fractions}, journal = {J. Number Theory}, volume = {174}, date = {2017}, pages = {369–383}, issn = {0022-314X}, review = {\MR{3597396}}} RohlinV. A.Exact endomorphisms of a lebesgue spaceRussianIzv. Akad. Nauk SSSR Ser. Mat.251961499–530ISSN 0373-2436Review MathReviews@article{R61, author = {Rohlin, V. A.}, title = {Exact endomorphisms of a Lebesgue space}, language = {Russian}, journal = {Izv. Akad. Nauk SSSR Ser. Mat.}, volume = {25}, date = {1961}, pages = {499–530}, issn = {0373-2436}, review = {\MR{0143873}}} WuChunLanTanFengHausdorff dimension of a chaotic set of shift in a countable symbolic spaceChinese, with English and Chinese summariesJ. South China Normal Univ. Natur. Sci. Ed.2007411–16ISSN 1000-5463Review MathReviews@article{WT07, author = {Wu, ChunLan}, author = {Tan, Feng}, title = {Hausdorff dimension of a chaotic set of shift in a countable symbolic space}, language = {Chinese, with English and Chinese summaries}, journal = {J. South China Normal Univ. Natur. Sci. Ed.}, date = {2007}, number = {4}, pages = {11–16}, issn = {1000-5463}, review = {\MR{2385049}}} WaltersPeterAn introduction to ergodic theoryGraduate Texts in Mathematics79Springer-Verlag, New York-Berlin1982ix+250ISBN 0-387-90599-5Review MathReviews@book{WP, author = {Walters, Peter}, title = {An introduction to ergodic theory}, series = {Graduate Texts in Mathematics}, volume = {79}, publisher = {Springer-Verlag, New York-Berlin}, date = {1982}, pages = {ix+250}, isbn = {0-387-90599-5}, review = {\MR{648108}}} WuJunA remark on the growth of the denominators of convergentsMonatsh. Math.14720063259–264ISSN 0026-9255Review MathReviews@article{Wu2006, author = {Wu, Jun}, title = {A remark on the growth of the denominators of convergents}, journal = {Monatsh. Math.}, volume = {147}, date = {2006}, number = {3}, pages = {259–264}, issn = {0026-9255}, review = {\MR{2215567}}} XiongJinChengYangZhongGuoChaos caused by a topologically mixing maptitle={Dynamical systems and related topics}, address={Nagoya}, date={1990}, series={Adv. Ser. Dynam. Systems}, volume={9}, publisher={World Sci. Publ., River Edge, NJ}, 1991550–572Review MathReviews@article{X91, author = {Xiong, JinCheng}, author = {Yang, ZhongGuo}, title = {Chaos caused by a topologically mixing map}, conference = {title={Dynamical systems and related topics}, address={Nagoya}, date={1990}, }, book = {series={Adv. Ser. Dynam. Systems}, volume={9}, publisher={World Sci. Publ., River Edge, NJ}, }, date = {1991}, pages = {550–572}, review = {\MR{1164918}}} XiongJinChengErratic time dependence of orbits for a topologically mixing mapEnglish, with Chinese summaryJ. China Univ. Sci. Tech.2119914387–396ISSN 0253-2778Review MathReviews@article{XX91, author = {Xiong, JinCheng}, title = {Erratic time dependence of orbits for a topologically mixing map}, language = {English, with Chinese summary}, journal = {J. China Univ. Sci. Tech.}, volume = {21}, date = {1991}, number = {4}, pages = {387–396}, issn = {0253-2778}, review = {\MR{1155316}}} XiongJinChengHausdorff dimension of a chaotic set of shift of a symbolic spaceSci. China Ser. A3819956696–708ISSN 1001-6511Review MathReviews@article{X95, author = {Xiong, JinCheng}, title = {Hausdorff dimension of a chaotic set of shift of a symbolic space}, journal = {Sci. China Ser. A}, volume = {38}, date = {1995}, number = {6}, pages = {696–708}, issn = {1001-6511}, review = {\MR{1351235}}}