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 \[ \limsup _{i\to +\infty }\rho (s_i x,s_iy)>0\ \text {and}\ \liminf _{i\to +\infty }\rho (s_ix,s_iy)=0. \]