Pinsker $\sigma$-algebra Character and mean Li-Yorke chaos

TL;DR

This paper establishes that for actions of infinite countable discrete amenable groups on compact spaces, the Pinsker σ-algebra acts as a characteristic factor, linking positive entropy to mean Li-Yorke chaos along specific sequences.

Contribution

It proves the Pinsker σ-algebra is a characteristic factor for group actions and connects positive entropy with mean Li-Yorke chaos in this context.

Findings

Pinsker σ-algebra is a characteristic factor for G-actions.

Positive topological entropy implies mean Li-Yorke chaos.

Results apply to sequences of finite subsets of G.

Abstract

Let $G$ be an infinite countable discrete amenable group. For any $G$-action on a compact metric space $X$, it is proved that for any sequence $(G_n)_{n\ge 1}$ consisting of non-empty finite subsets of $G$ with $\lim_{n\to \infty}|G_n|=\infty$, Pinsker $\sigma$-algebra is a characteristic factor for $(G_n)_{n\ge 1}$. As a consequence, for a class of $G$-topological dynamical systems, positive topological entropy implies mean Li-Yorke chaos along a class of sequences consisting of non-empty finite subsets of $G$.