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 $(X,\phi)$ over an invertible ergodic Polish system $(\Omega,\mathcal{F},\mathbb{P},\theta)$ admits a $\phi$-invariant random compact subset $K$ with $h_{top}(K,\phi)>0$, 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(\phi(a_i, \omega)x_1, \phi(a_i, \omega)x_2\big)=0,\quad\limsup_{N\to+\infty}\frac{1}{N}\sum_{i=1}^{N} d\big(\phi(a_i, \omega)x_1, \phi(a_i, \omega)x_2\big)>\epsilon(\omega), \end{align*} where $d$ is a compatible complete metric on $X$.