Uniform positive recursion frequency of any minimal dynamical system on a compact space

TL;DR

This paper proves that in minimal dynamical systems on compact spaces, the frequency of positive recurrence is uniformly bounded away from zero, extending to semiflows and amenable group actions.

Contribution

It establishes a uniform positive lower bound on recurrence frequencies for all points in minimal systems, generalizing previous results to broader classes of dynamical systems.

Findings

Uniform positive recurrence frequency in minimal systems

Extension to minimal semiflows and amenable group actions

Quantitative recurrence bounds for all points

Abstract

Using Gottschalk's notion\,---\,weakly locally almost periodic point, we show in this paper that if $f\colon X\rightarrow X$ is a minimal continuous transformation of a compact Hausdorff space $X$ to itself, then for all entourage $\varepsilon$ of $X$, \begin{equation*} \inf_{x\in X}\left\{\liminf_{N-M\to\infty}\frac{1}{N-M}\sum_{n=M}^{N-1}1_{\varepsilon[x]}(f^nx)\right\}>0. \end{equation*} An analogous assertion also holds for each minimal $C^0$-semiflow $\pi\colon \mathbb{R}_+\times X\rightarrow X$ and for any minimal transformation group with discrete amenable phase group.