Positive entropy implies chaos along any infinite sequence

Abstract

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