数学物理学报  2018, Vol. 38 Issue (3): 446-453   PDF    
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王建军
∑(X)上权移位算子的不变分布混沌性
王建军     
四川农业大学理学院 四川雅安 625014
摘要:该文证明对任意0<ε<2,赋范线性空间∑(X)上权移位算子Bw存在不可数不变分布ε-不规则集,进而推广了文献[27, 32]的主要结果.
关键词权移位算子    分布ε-混沌集    分布混沌集    
On the Invariance of Maximal Distributional Chaos of Weighted Shift Operators on ∑(X)
Wang Jianjun     
School of Sciences, Sichuan Agricultural University, Sichuan Ya'an 625014
Abstract: This paper proves that the weighted shift operator Bw on an normed linear space ∑(X) admits an invariant distributionally ε-scrambled set for any 0 < ε < 2, improving the main results of[27, 32].
Key words: Weighted shift operator     Wistributional ε-chaos     Distributionally scrambled set    
1 引言和基本定义

本文用 ${\Bbb N}$ ${\Bbb Z}^+$ 分别表示正整数与非负整数集.设 $(X, d)$ 为一度量空间, 若映射 $f:X\longrightarrow X$ 连续, 则称 $(X, f)$ 为一个系统.

众所周知, 混沌是指系统在沿着时间维度演化过程中, 其微观个体的状态相对于人们预测能力而言的不确定性, 它是系统复杂性的一个重要量度. 1975年, Li和Yorke[1]首次提出在区间上混沌的精确数学定义, 即大家熟知的Li-Yorke混沌.他们用点对的邻近和非渐进行为来刻画系统演化的复杂性.从此, 混沌的数学基础便成为一个新兴的研究方向, 受到广泛的关注和深入的研究.之后, 部分学者将Li-Yorke混沌的定义推广到更一般的度量空间, 如文献[2-3].

定义1.1[2-3]  设 $(X, f)$ 为一动力系统, $D\subset X$ $\delta>0$ .

(a) 称 $D$ 为系统 $(X, f)$ $\delta$ -不规则集, 如果对任意 $x, y\in D\ (x\neq y)$ , 恒有

$ \liminf\limits_{n\rightarrow \infty}d(f^{n}(x), f^{n}(y))=0\; \mbox{且}\; \limsup\limits_{n\rightarrow \infty}d(f^{n}(x), f^{n}(y))>\delta. $

(b) 称 $(X, f)$ 是Li-Yorke混沌的, 如果其存在一个不可数的 $0$ -不规则集.

(c) 称 $(x, y)\in X\times X$ $f$ 的一个Li-Yorke混沌对, 如果

$ \liminf\limits_{n\rightarrow \infty}d(f^{n}(x), f^{n}(y))=0\; \mbox{且}\; \limsup\limits_{n\rightarrow \infty}d(f^{n}(x), f^{n}(y))>0. $

随着理论研究的不断深入和实际问题的需要, 学者们先后给出一系列混沌新定义, 诸如: Devaney混沌[4], 稠混沌, 稠 $\delta$ -混沌[5], $\omega$ -混沌[6], 分布混沌[7], Li-Yorke敏感[8], 序列分布混沌[9]等.为了弥补Li-Yorke混沌的局限性和不足, Schweizer和Smítal[7]于1994年提出了一个更强的混沌定义, 即分布混沌(distributional chaos).文献[7, 10]不仅从轨道逼近时间集密度进一步揭示了隐藏Li-Yorke混沌内在的丰富多彩的复杂性, 同时证明紧区间或者双曲符号空间上的限制映射的分布混沌等价于正拓扑熵.直到2006年, Oprocha[10]在研究Devaney混沌与分布混沌关系时, 举例说明两者之间是互不蕴含的.

任取点对 $(x, y)\in X\times X$ $n\in{\Bbb N}$ , 令

$ \xi_n(t, x, y, f)=:\{k\in {\Bbb Z}^+:d(f^k(x), f^k(y))<t, 0\leq k< n\}. $

定义 $f$ 的下分布函数、上分布函数 ${\Bbb R}\longrightarrow [0, 1]$ 分别如下

$ F_{x, y}(t, f)=\liminf\limits_{n\to \infty} \frac{1}{n} \left|\xi_n(t, x, y, f)\right|, \;F_{x, y}^{*}(t, f)=\limsup\limits_{n\to \infty} \frac{1}{n} \left|\xi_n(t, x, y, f)\right|, $

其中 $|A|$ 表示集合 $A$ 的基数.显然, 分布函数 $F_{xy}(f)$ $F_{xy}^{*}(f)$ 都是非递减的函数.

定义1.2[7]  设 $(X, f)$ 为一动力系统, $D\subset X$ $\varepsilon>0$ .

(a) 称 $(x, y)\in X\times X$ $f$ 的一个分布 $\varepsilon$ -混沌对, 如果 $F_{x, y}^{*}(t, f)=1, \forall t>0$ 并且 $F_{x, y}(\varepsilon, f)=0$ .

(b) 称 $D$ 为分布 $\varepsilon$ -不规则集, 如果任意不同的点对 $(x, y)\in D\times D$ 都是分布 $\varepsilon$ -混沌对.

(c) 称系统 $(X, f)$ 是分布 $\varepsilon$ -混沌的, 如果存在不可数的分布 $\varepsilon$ -不规则集.

(d) 称系统 $(X, f)$ 是最大分布混沌的, 如果对任意 $0<\varepsilon<{\rm diam}(X)$ , $(X, f)$ 是分布 $\varepsilon$ -混沌的.

$(X, \|\cdot\|)$ 为复数域 ${\Bbb C}$ 上的赋范线性空间(非平凡且不一定完备), 置

$ \Sigma(X)=X^{{\Bbb Z}^+}=\{(x_0, x_1, x_{2}, \cdots):x_k\in X, k\in{\Bbb Z}^+\}. $

赋予 $\Sigma(X)$ 如下乘积拓扑度量 $\varrho$ :对任意 $x=(x_0, x_1, x_{2}, \cdots), y=(y_0, y_1, y_{2}, \cdots)\in \Sigma(X)$ ,

$ \varrho(x, y)=\sum\limits_{n=0}^{\infty}\frac{1}{2^n}\frac{\|x_n-y_n\|}{1+\|x_n-y_n\|}. $

显然, 在该度量意义下, $\Sigma(X)$ 的直径 ${\rm diam}(\Sigma(X))=2$ .定义 $\Sigma(X)$ 上加法" $+$ "和数乘法"·"运算如下:对任意 $x, y\in\Sigma(X)$ , $\alpha\in {\Bbb C}$ , $ (x+y)_n=x_n+y_n, \ (\alpha\cdot x)_n=\alpha x_n. $ 容易验证 $(\Sigma(X), \varrho)$ 为一个Fréchet空间.

对任意一组权值 $w=\{w_k:k\in {\Bbb Z}^+\}\subset {\Bbb C}\setminus\{0\}$ , 定义由权值 $w$ 诱导的权移位映射 $B_w:\Sigma(X)\longrightarrow\Sigma(X)$ 如下

$ B_w(x_0, x_1, x_2, \cdots)=(w_0x_1, w_1x_2, w_2x_3, \cdots). $

显然, $(\Sigma(X), B_w)$ 为一线性动力系统.任取 $n\in{\Bbb N}$ $(x_0, x_1, x_{2}, \cdots)\in\Sigma(X)$ , 不难验证

$ \begin{equation}\label{e1} B_w^n(x_0, x_1, x_2, \cdots)=\bigg(\bigg(\prod\limits_{s=0}^{n-1}w_s\bigg)x_n, \cdots, \bigg(\prod\limits_{s=0}^{n-1}w_{m+s}\bigg)x_{n+m}, \cdots\bigg). \end{equation} $ (1.1)

近年来, 关于无限维空间上线性算子混沌性引起了不少学者的关注; 尤其是Fréchet空间或者Banach空间上移位算子不同混沌性研究, 请参见文献[11-25]. 2009年, Fu等[16]根据 $p$ -不规则集, 极大不规则集和轨道不变性研究了权移位算子 $B_w$ 的Li-Yorke不规则集的结构.之后, 文献[26]继续研究了 $B_w$ 的Devaney混沌性、分布混沌性和Li-Yorke敏感性等.在此基础上, 文献[27-28]证明对任意 $0<\varepsilon<{\rm diam}(\Sigma(X))$ , $B_w$ 是分布 $\varepsilon$ -混的, 即 $B_w$ 是最大分布混沌的.另外, Wu等[29-30]系统研究了Köthe系列空间上权移位算子的Li-Yorke混沌性和分布混沌性, 得到了Li-Yorke和第二类型分布混沌的等价刻画, 同时得到了该算子为极大分布混沌的一个充分条件.

基于此, 受文献[31-32]的启发, 该文继续研究赋范线性空间 $\Sigma(X)$ 上权移位算子 $B_w$ 的极大不变分布混沌集.具体地讲, 证明存在不可数不变子集 $D\subset \Sigma(X)$ , 使得对任意 $0<\varepsilon<{\rm diam}(\Sigma(X))$ , $D$ 是分布 $\varepsilon$ -不规则集.

2 主要定理及其证明

本节发展文献[31-32]的方法证明权移位算子 $B_w$ 是极大分布混沌的, 从而推广和改进了文献[27, 32]的主要结果.

定理2.1  设 $B_w$ $\Sigma(X)$ 上权移位算子, 则存在不可数不变子集 $D\subset \Sigma(X)$ , 使得对任意 $0<\varepsilon<{\rm diam}(\Sigma(X))$ , $D$ 是分布 $\varepsilon$ -不规则集.

  设 $T_1={\Bbb T}_1=2$ , $T_n=2^{T_1+T_2+\cdots+T_{n-1}}$ ${\Bbb T}_n=\sum\limits_{i=1}^{n}T_i$ , $n>1$ .将所有奇素数按照序 $"<"$ 排列记为: $p_1, p_2, p_3, \cdots $ .

任取 $m, n\in {\Bbb N}$ , 令

$ \ \ \ \ \ \ \ \ {\Bbb A}=\{j\in{\Bbb N}:{\Bbb T}_{4n-1}\leq j<{\Bbb T}_{4n}\}, \ \ \ \ \ \ \ \ {\Bbb B}=\{j\in{\Bbb N}:{\Bbb T}_{4n-3}\leq j<{\Bbb T}_{4n-2}\}, \\ {\Bbb C}_{n, m}=\left\{j\in{\Bbb N}:{\Bbb T}_{p_n^m-1}+(2k+1)m\leq j<{\Bbb T}_{p_n^m-1}+2(k+1)m, 1\leq 2k+1\leq \left[\frac{T_{p_n^m}}{m}\right]-1\right\}, $

$ {\Bbb D}_{n, m}=\left\{j\in{\Bbb N}:{\Bbb T}_{p_n^m-1}+2km\leq j<{\Bbb T}_{p_n^m-1}+(2k+1)m, 0\leq 2k\leq \left[\frac{T_{p_n^m}}{m}\right]-1\right\}, $

其中, $\left[x\right]$ 表示 $x$ 的整数部分.

接下来构造不变不可数集 $D$ .根据文献[33, 引理5], 存在不可数子集 $E\subset \Sigma_2:=\{0, 1\}^{{\Bbb Z}^+}$ , 使得 $E$ 中任意不同点 $x=x_0x_1\cdots, y=y_0y_1\cdots$ 满足:对无限多个 $m\in {\Bbb N}$ , $x_m=y_m$ 且对无限多个 $n$ , $x_n\neq y_n$ .近而, 对任意 $x=x_0x_1\cdots\in E$ , 定义 $\varphi(x)=(\varphi(x)_0, \varphi(x)_1, \cdots)$ , 如下

$ \varphi(x)_j= \left\{\begin{array}{ll} x_0\cdot e,&{\rm } {j=0}, \\ 0,&{\rm } {j\in{\Bbb A}_n}, \\[2mm] \frac{(4n-2)\prod\limits_{s=0}^{{\Bbb T}_{4n-2}}(1+\left|w_s\right|)}{\prod\limits_{s=0}^{j-1}w_s}x_n\cdot e, ~~& {\rm } {j\in{\Bbb B}_n}, \\[7mm] \frac{P_n^m\prod\limits_{s=0}^{{\Bbb T}_{P_n^m}}(1+\left|w_s\right|)}{\prod\limits_{s=0}^{j-1}w_s}\cdot e,&{\rm } {j\in{\Bbb C}_{n, m}}, \\[3mm] 0,&{\rm } {j\in{\Bbb D}_{n, m}}, \\ 0,&{\rm } {\mbox{其它}}, \end{array}\right. $

其中, $e$ $X$ 的一个单位元.显然, $\varphi(E)\subset\Sigma(X)$ .令

$ D=\bigcup\limits_{n\in{\Bbb Z}^+}B_w^n(\varphi(E)). $

容易验证, $B_w(D)\subset D\subset \Sigma(X)$ $D$ 是不可数的.

任意取定 $D$ 中不同两个点 $\mu=\mu_0\mu_1\cdots, \tau=\tau_0\tau_1\cdots$ , 则存在 $a=a_0a_1\cdots, b=b_0b_1\cdots \in E$ $p, q\in {\Bbb Z}^+$ , 使得 $\mu=B_w(\varphi^p(a))$ , $\tau=B_w(\varphi^q(b))$ .

由(1.1)式知, 对任意 $j\in{\Bbb Z}^+$ ,

$ \begin{equation}\label{e2} B_w^j(\mu)=B_w^{j+p}(\varphi(a))=\bigg(\bigg(\prod\limits_{s=0}^{j+p-1}w_s\bigg)\varphi(a)_{j+p}, \cdots, \bigg(\prod\limits_{s=0}^{j+p-1}w_{i+s}\bigg)\varphi(a)_{j+p+i}, \cdots\bigg), \end{equation} $ (2.1)

并且

$ \begin{equation}\label{e3} B_w^j(\tau)=B_w^{j+q}(\varphi(b))=\bigg(\bigg(\prod\limits_{s=0}^{j+q-1}w_s\bigg)\varphi(b)_{j+q}, \cdots, \bigg(\prod\limits_{s=0}^{j+q-1}w_{i+s}\bigg)\varphi(b)_{j+q+i}, \cdots\bigg). \end{equation} $ (2.2)

不妨设 $p\leq q$ .为了证明对任意 $\varepsilon\in(0, 2)$ , $(\mu, \tau)$ 是分布 $\varepsilon$ -混沌对, 下面分两种情形讨论.

情形1 $a\neq b$ .

对任意 ${\Bbb T}_{4n-1}\leq j<{\Bbb T}_{4n}$ , 由定义知 $\varphi(a)_j=\varphi(b)_j=0$ .结合(2.1)和(2.2)式, 对任意 ${\Bbb T}_{4n-1}\leq j<{\Bbb T}_{4n}$ , 有

$ \begin{eqnarray*} \varrho(B_w^j(\mu), B_w^j(\tau))&=&\varrho(B_w^{j+p}(\varphi(a)), B_w^{j+q}(\varphi(b)))\\ &=&\sum\limits_{i=0}^\infty\frac{1}{2^i}\psi_i(a, b)\\ &\leq&\sum\limits_{i={\Bbb T}_{4n}-(j+q+1)}^\infty\frac{1}{2^i}\psi_i(a, b)\\ &\leq&\sum\limits_{i={\Bbb T}_{4n}-j-(q+1)}^\infty\frac{1}{2^i}\ \ \ (\mbox{由于}\ j\leq {\Bbb T}_{4n-1}+\frac{T_{4n}}{2}\leq T_{4n})\\ &\leq&\sum\limits_{i=\frac{T_{4n}}{2}-(q+1)}^\infty\frac{1}{2^i}\\ &=&\frac{1}{2^{\frac{T_{4n}}{2}-(q+1)}}\longrightarrow 0, \;(n\longrightarrow \infty), \end{eqnarray*} $

其中

$ \psi_i(a, b)=\frac{\bigg\|\bigg(\prod\limits_{s=0}^{j+p-1}w_{i+s}\bigg)\varphi(a)_{j+p+i} -\bigg(\prod\limits_{s=0}^{j+q-1}w_{i+s}\bigg) \varphi(b)_{j+q+i}\bigg\|}{1+\bigg\|\bigg(\prod\limits_{s=0}^{j+p-1}w_{i+s}\bigg) \varphi(a)_{j+p+i}-\bigg(\prod\limits_{s=0}^{j+q-1}w_{i+s}\bigg)\varphi(b)_{j+q+i}\bigg\|}. $

所以对任意 $t>0$ , 存在 $N_1\in{\Bbb N}$ , 使得对任意 $n>N_1$ 及任意 ${\Bbb T}_{4n-1}\leq j\leq{\Bbb T}_{4n-1}+\frac{T_{4n}}{2}$ ,

$ \varrho(B_w^j(\mu), B_w^j(\tau))<t. $

进而

$ \begin{eqnarray*} F_{\mu, \tau}^{*}(t, B_w)&=&\limsup\limits_{n\to \infty} \frac{1}{n}\left|\xi_n(t, \mu, \tau, B_w)\right|\\ &\geq&\limsup\limits_{n\to\infty} \frac{1}{{\Bbb T}_{4n-1}+\frac{T_{4n}}{2}}\left|\xi_{{\Bbb T}_{4n-1}+\frac{T_{4n}}{2}}(t, \mu, \tau, B_w)\right|\\ &\geq&\limsup\limits_{n\to\infty}\frac{\frac{T_{4n}}{2}}{{\Bbb T}_{4n-1}+\frac{T_{4n}}{2}}\\ &=&\limsup\limits_{n\to\infty}\frac{2^{{\Bbb T}_{4n-1}-1}}{{\Bbb T}_{4n-1}+2^{{\Bbb T}_{4n-1}-1}}\\ &=& 1. \end{eqnarray*} $

另一方面, 由于 $a\neq b$ , 因此存在无限序列 $\{n_i\}_{i=1}^{\infty}\subset{\Bbb N}$ 使得 $a_{n_i}\neq b_{n_i}$ .结合(2.1)和(2.2)式知, 对任意 ${\Bbb T}_{4n_i-3}\leq j<{\Bbb T}_{4n_i-2}-q$ 及任意 $0\leq k\leq \frac{T_{4n_i-2}}{2}-(q+1)$ , 由于

$ \begin{eqnarray*} \psi_k(a, b)&=&\frac{\bigg\|\bigg(\prod\limits_{s=0}^{j+p-1}w_{k+s}\bigg)\varphi(a)_{j+p+k}-\bigg(\prod\limits_{s=0}^{j+q-1}w_{k+s}\bigg)\varphi(b)_{j+q+k}\bigg\|}{1+ \bigg\|\bigg(\prod\limits_{s=0}^{j+p-1}w_{k+s}\bigg)\varphi(a)_{j+p+k}-\bigg(\prod\limits_{s=0}^{j+q-1}w_{k+s}\bigg)\varphi(b)_{j+q+k}\bigg\|}\\ &=&\frac{\bigg\|\frac{(4n_i-2)\prod\limits_{s=0}^{{\Bbb T}_{4n_i-2}}(1+\left|w_s\right|)}{\prod\limits_{s=0}^{k-1}w_{s}}a_{n_i}-\frac{(4n_i-2)\prod\limits_{s=0}^{{\Bbb T}_{4n_i-2}}(1+\left|w_s\right|)}{\prod\limits_{s=0}^{k-1}w_{s}}b_{n_i}\bigg\|} {1+\bigg\|\frac{(4n_i-2)\prod\limits_{s=0}^{{\Bbb T}_{4n_i-2}}(1+\left|w_s\right|)}{\prod\limits_{s=0}^{k-1}w_{s}}a_{n_i}-\frac{(4n_i-2)\prod\limits_{s=0}^{{\Bbb T}_{4n_i-2}}(1+\left|w_s\right|)}{\prod\limits_{s=0}^{k-1}w_{s}}b_{n_i}\bigg\|}\\ &\geq&\frac{(4n_i-2)\bigg\|a_{n_i}-b_{n_i}\bigg\|}{1+(4n_i-2)\bigg\|a_{n_i}-b_{n_i}\bigg\|}\\ &=&\frac{4n_i-2}{1+(4n_i-2)}, \end{eqnarray*} $

则有

$ \begin{eqnarray*} \varrho(B_w^j(\mu), B_w^j(\tau))&=&\varrho(B_w^{j+p}(\varphi(a)), B_w^{j+q}(\varphi(b)))\\ &=&\sum\limits_{k=0}^\infty\frac{1}{2^k}\psi_k(a, b)\\ &\geq&\sum\limits_{k=0}^{\frac{T_{4n_i-2}}{2}-(q+1)}\frac{1}{2^k}\psi_k(a, b)\\ &\geq&\frac{4n_i-2}{1+(4n_i-2)}\cdot\sum\limits_{k=0}^{\frac{T_{4n_i-2}}{2}-(q+1)}\frac{1}{2^k}\longrightarrow 2, \; (i\longrightarrow \infty). \end{eqnarray*} $

所以, 存在 $N_2\in {\Bbb N}$ , 使得对任意 $i\geq N_2$ 及任意 ${\Bbb T}_{4n_i-3}\leq j\leq{\Bbb T}_{4n_i-3}+\frac{T_{4n_i-2}}{2}-(q+1)$ ,

$ \varrho(B_w^j(\mu), B_w^j(\tau))\geq \varepsilon. $

因此, 对任意 $0<\varepsilon<2$ , 有

$ \begin{eqnarray*} F_{\mu, \tau}(\varepsilon, B_w)&=&\liminf\limits_{n\to \infty} \frac{1}{n}\left|\xi_n(t, \mu, \tau, B_w)\right|\\ &\leq&\liminf\limits_{n\to\infty} \frac{1}{{\Bbb T}_{4n_i-3}+\frac{T_{4n_i-2}}{2}-(q+1)}\left|\xi_{{\Bbb T}_{4n_i-3}+\frac{T_{4n_i-2}}{2}-(q+1)}(\varepsilon, \mu, \tau, B_w)\right|\\ &\leq&\liminf\limits_{n\to\infty}\frac{{\Bbb T}_{4n_i-3}}{{\Bbb T}_{4n_i-3}+\frac{T_{4n_i-2}}{2}-(q+1)}\\ &\leq&\liminf\limits_{n\to\infty}\frac{{\Bbb T}_{4n_i-3}}{{\Bbb T}_{4n_i-3}+2^{{\Bbb T}_{4n_i-3}-1}-(q+1)}\\ &=& 0. \end{eqnarray*} $

情形2 $a=b$ .

同样不妨设 $p<q$ .根据情形1的讨论, 对任意 $t>0$ , 显然

$ F_{\mu, \tau}^*(t, B_w)=1. $

另一方面, 由于 $a=b$ , 因此对任意 ${\Bbb T}_{P_n^{q-p}-1}\leq j+p<{\Bbb T}_{P_n^{q-p}-1}+\frac{T_{P_n^{q-p}}}{2}-(q-p)$ 及任意 $0\leq k\leq \frac{T_{P_n^{q-p}}}{2}-(q+1)$ , 容易验证

$ \begin{eqnarray*} &&\bigg\|\bigg(\prod\limits_{s=0}^{j+p-1}w_{k+s}\bigg)\varphi(a)_{j+p+k}-\bigg(\prod\limits_{s=0}^{j+p+(q-p)-1} w_{k+s}\bigg)\varphi(a)_{j+p+(q-p)+k}\bigg\|\\ &=&\left|\frac{P_n^{q-p}\prod\limits_{s=0}^{{\Bbb T}_{P_n^{q-p}}}(1+\left|w_s\right|)} {\prod\limits_{s=0}^{k-1}w_{s}}\right|\geq P_n^{q-p}. \end{eqnarray*} $

由此可知

$ \begin{eqnarray*} \psi_k(a, b)&=&\frac{\bigg\|\bigg(\prod\limits_{s=0}^{j+p-1}w_{k+s}\bigg)\varphi(a)_{j+p+k}-\bigg(\prod\limits_{s=0}^{j+p+(q-p)-1}w_{k+s}\bigg)\varphi(a)_{j+p+(q-p)+k}\bigg\|}{1+ \bigg\|\bigg(\prod\limits_{s=0}^{j+p-1}w_{k+s}\bigg)\varphi(a)_{j+p+k}-\bigg(\prod\limits_{s=0}^{j+p+(q-p)-1}w_{k+s}\bigg)\varphi(a)_{j+p+(q-p)+k}\bigg\|}\\ &\geq&\frac{P_n^{q-p}}{1+P_n^{q-p}}. \end{eqnarray*} $

从而

$ \begin{eqnarray*} \varrho(B_w^j(\mu), B_w^j(\tau))&=&\varrho(B_w^{j+p}(\varphi(a)), B_w^{j+p+(q-p)}(\varphi(b)))\\ &=&\sum\limits_{k=0}^\infty\frac{1}{2^k}\psi_k(a, b)\\ &\geq&\sum\limits_{k=0}^{\frac{T_{P_n^{q-p}}}{2}-(q+1)}\frac{1}{2^k}\psi_k(a, b)\\ &\geq&\frac{P_n^{q-p}}{1+P_n^{q-p}}\cdot\sum\limits_{k=0}^{\frac{T_{P_n^{q-p}}}{2}-(q+1)}\frac{1}{2^k}\longrightarrow 2, \; (n\longrightarrow \infty). \end{eqnarray*} $

所以, 存在 $N_3\in {\Bbb N}$ , 使得对任意 $n\geq N_3$ 及任意 ${\Bbb T}_{P_n^{q-p}-1}\leq j\leq{\Bbb T}_{P_n^{q-p}-1}+\frac{T_{P_n^{q-p}}}{2}-(q+1)$ ,

$ \varrho(B_w^j(\mu), B_w^j(\tau))\geq \varepsilon. $

因此, 对任意 $0<\varepsilon<2$ , 有

$ \begin{eqnarray*} F_{\mu, \tau}(\varepsilon, B_w)&=&\liminf\limits_{n\to \infty} \frac{1}{n}\left|\xi_n(t, \mu, \tau, B_w)\right|\\ &\leq&\liminf\limits_{n\to\infty} \frac{1}{{\Bbb T}_{P_n^{q-p}-1}+\frac{T_{P_n^{q-p}}}{2}-(q+1)}\left|\xi_{{\Bbb T}_{P_n^{q-p}-1}+\frac{T_{P_n^{q-p}}}{2}-(q+1)}(\varepsilon, \mu, \tau, B_w)\right|\\ &\leq&\liminf\limits_{n\to\infty}\frac{{\Bbb T}_{P_n^{q-p}-1}}{{\Bbb T}_{P_n^{q-p}-1}+\frac{T_{P_n^{q-p}}}{2}-(q+1)}\\ &\leq&\liminf\limits_{n\to\infty}\frac{{\Bbb T}_{P_n^{q-p}-1}}{{\Bbb T}_{P_n^{q-p}-1}+2^{{\Bbb T}_{P_n^{q-p}-1}-1}-(q+1)}\\ &=& 0. \end{eqnarray*} $

根据情形1和情形2的讨论以及 $\mu, \tau$ 的任意性知, 对任意 $0<\varepsilon<{\rm diam}\Sigma(X)$ , $D$ 是分布 $\varepsilon$ -混沌集.

参考文献
[1] Li T Y, Yorke J A. Period three implies chaos. Amer Math Monthly, 1975, 82(10): 985–992. DOI:10.1080/00029890.1975.11994008
[2] Oprocha P. Relations between distributional and Devaney chaos. Chaos, 2006, 16: 033112. DOI:10.1063/1.2225513
[3] Huang W, Ye X D. Devaney chaos or 2-scattering implies Li-Yorke chaos. Topo Appl, 2002, 117(3): 259–272. DOI:10.1016/S0166-8641(01)00025-6
[4] Devaney R L. An Introduction to Chaotic Dynamical Systems (2nd ed). Redwood City, CA: AddisonWesley Publishing Company, 1989
[5] Snoha L. Dense chaos. Comment Math Univ Carolin, 1992, 33(4): 747–752.
[6] Li S H. ω-Chaos and topological entropy. Trans Amer Math Soc, 1993, 399(1): 243–249.
[7] Schweizer B, Smítal J. Measures of chaos and a spectral decomposition of dynamical systems on the interval. Trans Amer Math Soc, 1994, 344: 737–754. DOI:10.1090/tran/1994-344-02
[8] Akin E, Kolyada S. Li-Yorke sensitivity. Nonlinearity, 2003, 16: 1421–1433. DOI:10.1088/0951-7715/16/4/313
[9] Wang L. D, Huang G F, Huan S M. Distributional chaos in sequence. Nonlinear Anal, 2007, 67(7): 2131–2136. DOI:10.1016/j.na.2006.09.005
[10] Oprocha P, Wilczyński P. Shift spaces and distributional chaos. Chaos Solitons and Fractals, 2007, 31(2): 347–355. DOI:10.1016/j.chaos.2005.09.069
[11] Bermúdez T, Bonilla A, Martínez-Giménez F, Peris A. Li-Yorke and distributionally chaotic operators. J Math Anal Appl, 2011, 373: 83–93. DOI:10.1016/j.jmaa.2010.06.011
[12] Bayart F, Grivaux S. Frequently hypercyclic operators. Trans Amer Math Soc, 2006, 358: 5083–5117. DOI:10.1090/S0002-9947-06-04019-0
[13] Bès J, Peris A. Hereditarily hypercyclic operators. J Funct Anal, 1999, 167: 94–112. DOI:10.1006/jfan.1999.3437
[14] Bès J, Peris A. Disjointness in hypercyclicity. J Math Anal Appl, 2007, 336: 297–315. DOI:10.1016/j.jmaa.2007.02.043
[15] Chan K C, Shapiro J H. The cyclic behavior of translation operators on Hilbert spaces of entire functions. Indiana Univ Math J, 1991, 40: 1421–1449. DOI:10.1512/iumj.1991.40.40064
[16] Fu X C, You Y C. Chaotic sets of shift and weighted shift maps. Nonlin Anal, 2009, 71: 2141–2152. DOI:10.1016/j.na.2009.01.049
[17] Grosse-Erdmann K G. Hypercyclic and chaotic weighted shifts. Studia Math, 2000, 139: 47–68. DOI:10.4064/sm-139-1-47-68
[18] Grosse-Erdmann K G, Peris A. Frequently dense orbits. C R Math Acad Sci Paris, 2005, 341: 123–128. DOI:10.1016/j.crma.2005.05.025
[19] Martínez-Giménez F, Peris A. Chaos for backward shift operators. Int J Bifurcation and Chaos, 2002, 12: 1703–1715. DOI:10.1142/S0218127402005418
[20] Martínez-Giménez F. Chaos for power series of backward shift operators. Proc Amer Math Soc, 2007, 135: 1741–1752. DOI:10.1090/S0002-9939-07-08658-3
[21] Salas H N. Hypercyclic weighted shifts. Trans Amer Math Soc, 1995, 347: 993–1004. DOI:10.1090/tran/1995-347-03
[22] Wu X X, Zhu P Y. The principal measure of a quantum harmonic oscillator. J Phys A:Math Theor, 2011, 44: 505101. DOI:10.1088/1751-8113/44/50/505101
[23] Wu X X, Zhu P Y. On the equivalence of four chaotic operators. Appl Math Lett, 2012, 25: 545–549. DOI:10.1016/j.aml.2011.09.055
[24] 吴新星, 王建军. 关于P-极小动力系统的一些注记. 数学物理学报, 2016, 36A(5): 879–885.
Wu X X, Wang J J. Some remarks on P-minimal dynamical systems. Acta Math Scientia, 2016, 36A(5): 879–885.
[25] 吴新星. 关于弱specification性质的一个注记. 数学物理学报, 2017, 37A(4): 601–606.
Wu X X. A remark on the weak specification property. Acta Math Scientia, 2017, 37A(4): 601–606.
[26] Wu X X, Zhu P Y. Chaos in a class of nonconstant weighted shift operation. Int J Bifurcation and Chaos, 2013, 23(1): 1350010. DOI:10.1142/S0218127413500107
[27] Wu X X, Zhu P Y, Lu T X. Uniform distributional chaos for weighted shift operators. Applied Mathematics Letters, 2013, 26: 130–133. DOI:10.1016/j.aml.2012.04.008
[28] 卢天秀, 朱培勇, 吴新星. ∑(X)上移位算子的一致分布混沌和准测度. 应用数学学报, 2015, 38A(1): 1–7.
Lu T X, Zhu P Y, Wu X X. Principal measure and uniform distributional chaos of weighted shift operators on ∑(X). Acta Math Appl Sinica, 2015, 38A(1): 1–7.
[29] Wu X X, Chen G R, Zhu P Y. Invariance of chaos from backward shift on the Köthe sequence space. Nonlinearity, 2014, 27: 271–288. DOI:10.1088/0951-7715/27/2/271
[30] Wu X X, Wang L D, Chen G R. Weighted backward shift operators with invariant distributionally scrambled subsets. Ann Funct Anal, 2017, 8: 199–210. DOI:10.1215/20088752-3802705
[31] Wu X X, Chen G R. On the invariance of maximal distributional chaos under an annihilation operator. Applied Mathematics Letters, 2013, 26: 1134–1140. DOI:10.1016/j.aml.2013.06.011
[32] Wu X X, Zhu P Y. Invariant scrambled sets and maximal distributional chaos. Annales Polonici Mathematici, 2013, 109(3): 271–278. DOI:10.4064/ap109-3-3
[33] Liao G F, Fan Q J. Minimal subshifts which display Schweizer-Smital chaos and have zero topological entropy. Science in China, 1998, 41: 33–38. DOI:10.1007/BF02900769