# 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 and Continuous Dynamical Systems, 2016, 36 (6) : 2931-2944. doi: 10.3934/dcds.2016.36.2931
##### References:
 [1] J. Egawa, A characterization of regularly almost periodic minimal flows, Proc. Japan Acad. Ser. A, 71 (1995), 225-228. doi: 10.3792/pjaa.71.225. [2] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, New Jersey, 1981. [3] W. H. Gottschalk, Almost periodic points with respect to transformation semi-groups, Annals of Math., 47 (1946), 762-766. doi: 10.2307/1969233. [4] W. H. Gottschalk, A survey of minimal sets, Ann. Inst. Fourier, Grenoble, 14 (1964), 53-60. doi: 10.5802/aif.160. [5] W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, Amer. Math. Soc. Coll. Publ., Vol. 36, Amer. Math. Soc., Providence, R.I., 1955. [6] A. Miller and J. Rosenblatt, Characterizations of regular almost periodicity in compact minimal abelian flows, Trans. Amer. Math. Soc., 356 (2004), 4909-4929. doi: 10.1090/S0002-9947-04-03538-X. [7] D. Montgomery, Almost periodic transformation groups, Trans. Amer. Math. Soc., 42 (1937), 322-332. doi: 10.1090/S0002-9947-1937-1501924-0. [8] V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, New Jersey 1960. [9] A. Weil, L'Integration Dans Les Groupes Topologiques et Ses Applications, Actualitiés scientifiques, No. 869, Paris, Hermann, 1938.

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##### References:
 [1] J. Egawa, A characterization of regularly almost periodic minimal flows, Proc. Japan Acad. Ser. A, 71 (1995), 225-228. doi: 10.3792/pjaa.71.225. [2] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, New Jersey, 1981. [3] W. H. Gottschalk, Almost periodic points with respect to transformation semi-groups, Annals of Math., 47 (1946), 762-766. doi: 10.2307/1969233. [4] W. H. Gottschalk, A survey of minimal sets, Ann. Inst. Fourier, Grenoble, 14 (1964), 53-60. doi: 10.5802/aif.160. [5] W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, Amer. Math. Soc. Coll. Publ., Vol. 36, Amer. Math. Soc., Providence, R.I., 1955. [6] A. Miller and J. Rosenblatt, Characterizations of regular almost periodicity in compact minimal abelian flows, Trans. Amer. Math. Soc., 356 (2004), 4909-4929. doi: 10.1090/S0002-9947-04-03538-X. [7] D. Montgomery, Almost periodic transformation groups, Trans. Amer. Math. Soc., 42 (1937), 322-332. doi: 10.1090/S0002-9947-1937-1501924-0. [8] V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, New Jersey 1960. [9] A. Weil, L'Integration Dans Les Groupes Topologiques et Ses Applications, Actualitiés scientifiques, No. 869, Paris, Hermann, 1938.
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