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

Let $G$ be a countably infinite discrete group and let $ π:(X,μ,G)\to(Y,ν,G) $ be a nontrivial relatively mixing extension, where $X$ is a compact metrizable $G$-space. We prove that there exists a constant $δ>0$ such that, for every injective sequence $(s_i)_{i\geq 1}$ in $G$, there is a Cantor set