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On uniformly recurrent motions of topological semigroup actions

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  • 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.
    Mathematics Subject Classification: 37B05, 37B20.


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