Mean Li-Yorke chaos along some good sequences

TL;DR

This paper demonstrates that topological dynamical systems with positive entropy exhibit multivariant mean Li-Yorke chaos along certain sequences, including primes and generalized polynomials, revealing complex chaotic behavior in these systems.

Contribution

It establishes the presence of mean Li-Yorke chaos along 'good' sequences in systems with positive entropy, extending understanding of chaos in dynamical systems.

Findings

Chaos occurs along sequences like primes and generalized polynomials.

Existence of a Cantor set with specific mean distance properties.

Positive entropy implies multivariant mean Li-Yorke chaos along 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 $n\ge2$ and pairwise distinct points $x_1,x_2,\dotsc,x_n$ 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.