论文标题
无限二维随机动力学系统的任何无限序列沿任何无限序列的平均混乱
Mean Li-Yorke chaos along any infinite sequence for infinite-dimensional random dynamical systems
论文作者
论文摘要
在本文中，我们研究了无限二二维随机动力学系统的任何无限正整数序列的平均李 - 尤克混沌现象。确切地说，我们证明，如果在可逆的ergodic polish System $（ω，\ Mathcal {f}，\ mathbb {p}，θ，θ），$ -Intriant Compact $ -Invar $ K $ k_ $ k_ $ h_带有$ \ lim_ {i \ lim_ {i \ to+\\ to+\ ftty} a_i =+\ iffty $，用于$ \ mathbb {p} $ {p} $ - {p} $ - a，一个正整数序列$ \ mathbf {a} = \ {a_i \} _ {i \ in \ mathbb {n}} $带有$ \ lim_ {i \ to+\ to+\ fos+\ fos+\ infty} a_i =+\++\ iff infty $ $ω\inΩ$存在一个无数的子集$ s（ω）\ subset k（ω）$和$ε（ω）> 0 $，因此对于任何不同的点$ x_1 $，$ x_2 $，$ x_2 \ in S（ω）$，带有以下属性 \ begin {align*} \ liminf_ {n \ to+\ infty} \ frac {1} {N} {n} \ sum_ {i = 1} ω）x_2 \ big）= 0，\ quad \ limsup_ {n \ to+\ infty} \ frac {1} {n} {n} \ sum_ {i = 1}^{n} n} d \ big（ϕ（A_i，ω）x_1，ω） \ end {align*}其中$ d $是$ x $上兼容的完整度量。
In this paper, we study the mean Li-Yorke chaotic phenomenon along any infinite positive integer sequence for infinite-dimensional random dynamical systems. To be precise, we prove that if an injective continuous infinite-dimensional random dynamical system $(X,ϕ)$ over an invertible ergodic Polish system $(Ω,\mathcal{F},\mathbb{P},θ)$ admits a $ϕ$-invariant random compact subset $K$ with $h_{top}(K,ϕ)>0$, then given a positive integer sequence $\mathbf{a}=\{a_i\}_{i\in\mathbb{N}}$ with $\lim_{i\to+\infty}a_i=+\infty$, for $\mathbb{P}$-a.s. $ω\inΩ$ there exists an uncountable subset $S(ω)\subset K(ω)$ and $ε(ω)>0$ such that for any distinct points $x_1$, $x_2\in S(ω)$ with following properties \begin{align*} \liminf_{N\to+\infty}\frac{1}{N}\sum_{i=1}^{N} d\big(ϕ(a_i, ω)x_1, ϕ(a_i, ω)x_2\big)=0,\quad\limsup_{N\to+\infty}\frac{1}{N}\sum_{i=1}^{N} d\big(ϕ(a_i, ω)x_1, ϕ(a_i, ω)x_2\big)>ε(ω), \end{align*} where $d$ is a compatible complete metric on $X$.
