Mean Li-Yorke chaos along some good sequences

Abstract

If a topological dynamical system $(X,T)$ has positive topological entropy, then it is multivariant mean Li-Yorke chaotic along a sequence $\{a_k{\}}_{k=1}^{\infty }$ of positive integers which is "good" for pointwise ergodic convergence with a mild condition; more specifically, there exists a Cantor subset $K$ of $X$ such that for every $n\ge 2$ and pairwise distinct points $x_1,x_2,\dots ,x_n$ in $K$ we have \[\liminf_{N\to\infty}\frac{1}{N}\sum_{k=1}^N\max_{1\leq i<j\leq n} d(T^{a_k}x_i,T^{a_k}x_j)=0\] and \[\limsup_{N\to\infty}\frac{1}{N}\sum_{k=1}^N\min_{1\leq i<j\leq n} d(T^{a_k}x_i,T^{a_k}x_j)>0.\] Examples are given for the classic sequences of primes and generalized polynomials.

18 pages

References in corpus (1)

• Stable sets and mean Li-Yorke chaos in positive entropy actions of bi-orderable amenable groups

Cited by in corpus (4)

• Pinsker $\sigma$-algebra Character and mean Li-Yorke chaos
• $\Delta$-weakly mixing subsets along a collection of sequences of integers
• Directional Pinsker algebra and its applications
• Mean Li-Yorke chaos along polynomials of several variables and prime numbers