Positive entropy implies chaos along any infinite sequence

TL;DR

This paper proves that positive topological entropy in actions of infinite countable amenable groups guarantees Li-Yorke chaos along any infinite sequence, revealing a deep link between entropy and chaos in dynamical systems.

Contribution

It establishes that positive entropy implies Li-Yorke chaos along any sequence for group actions, extending chaos results to a broad class of group actions.

Findings

Positive entropy implies Li-Yorke chaos along any sequence.

Li-Yorke chaos exists on a Cantor set for such actions.

Chaos is guaranteed regardless of the chosen sequence in the group.

Abstract

Let $G$ be an infinite countable discrete amenable group. For any $G$-action on a compact metric space $(X,\rho)$, it turns out that if the action has positive topological entropy, then for any sequence $\{s_i\}_{i=1}^{+\infty}$ with pairwise distinct elements in $G$ there exists a Cantor subset $K$ of $X$ which is Li-Yorke chaotic along this sequence, that is, for any two distinct points $x,y\in K$, one has \[\limsup_{i\to+\infty}\rho(s_i x,s_iy)>0,\ \text{and}\ \liminf_{i\to+\infty}\rho(s_ix,s_iy)=0.\]