Li--Yorke Chaos Along Any Infinite Sequence: Relative Mixing, Sofic and Rokhlin Entropy versus Naive Entropy

Abstract

Let $G$ be a countably infinite discrete group and let $\pi :(X,\mu ,G)\to (Y,\nu ,G)$ be a nontrivial relatively mixing extension, where $X$ is a compact metrizable $G$-space. We prove that there exists a constant $\delta >0$ such that, for every injective sequence $(s_i{)}_{i\ge 1}$ in $G$, there is a Cantor set $K_{(s_i)}\subseteq X$ whose distinct points $x,x^{\prime }$ satisfy \[ \liminf_{i\to\infty}ρ(s_i x,s_i x')=0, \qquad \limsup_{i\to\infty}ρ(s_i x,s_i x')>δ. \] The method also yields higher-order scrambled Cantor sets. As a principal application, for a sofic group $G$, positive topological sofic entropy implies the preceding conclusion, answering a question of Huang, Li, and Ye. The same conclusion also holds for actions of arbitrary countably infinite discrete groups admitting an essentially free invariant measure of positive Rokhlin entropy. In contrast, we construct a hereditary subshift over a residually finite group with positive naive topological entropy for which all points converge to a common fixed point along a prescribed injective sequence. This answers a question of García-Ramos and Li negatively even for sofic groups.

arXiv commentThe revised version includes a counterexample showing that positive naive topological entropy is insufficient to guarantee Li--Yorke chaos along every prescribed infinite sequence. We welcome any comments, suggestions, or discussion regarding our manuscript