Ergodic measures in minimal group actions with finite topological sequence entropy

TL;DR

This paper investigates the relationship between topological sequence entropy and ergodic measures in minimal group actions, establishing bounds and structural properties especially for abelian groups.

Contribution

It proves a lower bound for the supremum of topological sequence entropy in minimal G-systems and characterizes the measure-theoretic structure for abelian groups.

Findings

Supremum of topological sequence entropy is bounded below by a logarithmic sum over ergodic measures.

For abelian groups, the maximal equicontinuous factor has a measure concentrated on fibers of constant size.

The paper links topological entropy with measure-theoretic properties in minimal group actions.

Abstract

Let $G$ be an infinite discrete countable group and $(X,G)$ be a minimal $G$-system. In this paper, we prove the supremum of topological sequence entropy of $(X,G)$ is not less than $\log(\sum_{\mu\in\mathcal{M}^e(X,G)}e^{h_\mu^*(X,G)})$. If additionally $G$ is abelian then there is a constant $K\in\mathbb{N}\cup\{\infty\}$ with $\log K\le h_{top}^*(X,G)$ such that $\nu(\{y\in H:|\pi^{-1}(y)|=K\})=1$ where $(H,G)$ is the maximal equicontinuous factor of $(X,G)$, $\pi:(X,G)\to (H,G)$ is the factor map and $\nu$ is the Haar measure of $H$.