On uniformly recurrent motions of topological semigroup actions

Bin  Chen; Xiongping Dai

doi:10.3934/dcds.2016.36.2931

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2016, Volume 36, Issue 6: 2931-2944. Doi: 10.3934/dcds.2016.36.2931

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On uniformly recurrent motions of topological semigroup actions

Bin Chen^1, and
Xiongping Dai^2,

1.
Department of Mathematics, Nanjing University, Nanjing 210093
2.
Department of Mathematics, Nanjing University, Nanjing, 210093

Received: September 30, 2014

Revised: September 30, 2015

Published: May 2017

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Abstract

Let G ↷ X be a topological action of a topological semigroup $G$ on a compact metric space $X$. We show in this paper that for any given point $x$ in $X$, the following two properties that both approximate to periodicity are equivalent to each other:
$\bullet$ For any neighborhood $U$ of $x$, the return times set $\{g\in G ： gx\in U\}$ is syndetic of Furstenburg in $G$.
$\bullet$ Given any $\varepsilon>0$, there exists a finite subset $K$ of $G$ such that for each $g$ in $G$, the $\varepsilon$-neighborhood of the orbit-arc $K[gx]$ contains the entire orbit $G[x]$.
This is a generalization of a classical theorem of Birkhoff for the case where $G=\mathbb{R}$ or $\mathbb{Z}$. In addition, a counterexample is constructed to this statement, while $X$ is merely a complete but not locally compact metric space.

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Mathematics Subject Classification: 37B05, 37B20.

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References

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