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

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 $(s_i)_{i\geq 1}$ in $G$, there is a Cantor set $K_{(s_i)}\subseteq X$ whose distinct points $x,x'$ satisfy \[ \liminf_{i\to\infty}\rho(s_i x,s_i x')=0, \qquad \limsup_{i\to\infty}\rho(s_i x,s_i x')>\delta. \] 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.