# American Institute of Mathematical Sciences

June  2016, 36(6): 2931-2944. doi: 10.3934/dcds.2016.36.2931

## On uniformly recurrent motions of topological semigroup actions

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

Received  October 2014 Revised  October 2015 Published  December 2015

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.
Citation: Bin Chen, Xiongping Dai. On uniformly recurrent motions of topological semigroup actions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 2931-2944. doi: 10.3934/dcds.2016.36.2931
##### References:
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##### References:
 [1] J. Egawa, A characterization of regularly almost periodic minimal flows,, Proc. Japan Acad. Ser. A, 71 (1995), 225.  doi: 10.3792/pjaa.71.225.  Google Scholar [2] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory,, Princeton University Press, (1981).   Google Scholar [3] W. H. Gottschalk, Almost periodic points with respect to transformation semi-groups,, Annals of Math., 47 (1946), 762.  doi: 10.2307/1969233.  Google Scholar [4] W. H. Gottschalk, A survey of minimal sets,, Ann. Inst. Fourier, 14 (1964), 53.  doi: 10.5802/aif.160.  Google Scholar [5] W. H. Gottschalk and G. A. Hedlund, Topological Dynamics,, Amer. Math. Soc. Coll. Publ., (1955).   Google Scholar [6] A. Miller and J. Rosenblatt, Characterizations of regular almost periodicity in compact minimal abelian flows,, Trans. Amer. Math. Soc., 356 (2004), 4909.  doi: 10.1090/S0002-9947-04-03538-X.  Google Scholar [7] D. Montgomery, Almost periodic transformation groups,, Trans. Amer. Math. Soc., 42 (1937), 322.  doi: 10.1090/S0002-9947-1937-1501924-0.  Google Scholar [8] V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations,, Princeton University Press, (1960).   Google Scholar [9] A. Weil, L'Integration Dans Les Groupes Topologiques et Ses Applications,, Actualitiés scientifiques, (1938).   Google Scholar
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