Positive entropy implies chaos along any infinite sequence

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 \[\limsup_{i\to+\infty}ρ(s_i x,s_iy)>0,\ \text{and}\ \liminf_{i\to+\infty}ρ(s_ix,s_iy)=0.\]

16 pages. This paper is dedicated to the memory of Anatoly Mikhailovich Stepin