Mean Li-Yorke chaos along any infinite sequence for infinite-dimensional random dynamical systems

Abstract

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 $(X,\varphi )$ over an invertible ergodic Polish system $(\Omega ,\mathcal{F},\mathbb{P},\theta )$ admits a $\varphi$-invariant random compact subset $K$ with $h_{top}(K,\varphi )>0$, 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. $\omega \in \Omega$ there exists an uncountable subset $S(\omega )\subset K(\omega )$ and $\epsilon (\omega )>0$ such that for any distinct points $x_1$, $x_2\in S(\omega )$ 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 $X$.

22 pages