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

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.