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

On uniformly recurrent motions of topological semigroup actions

Bin Chen 1, and  Xiongping Dai 2,

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.
Keywords: uniformly recurrent point, almost periodic point., $G$-space.
Mathematics Subject Classification: 37B05, 37B2.
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:
[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

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W. H. Gottschalk, Almost periodic points with respect to transformation semi-groups,, Annals of Math., 47 (1946), 762.  doi: 10.2307/1969233.  Google Scholar

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W. H. Gottschalk, A survey of minimal sets,, Ann. Inst. Fourier, 14 (1964), 53.  doi: 10.5802/aif.160.  Google Scholar

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W. H. Gottschalk and G. A. Hedlund, Topological Dynamics,, Amer. Math. Soc. Coll. Publ., (1955).   Google Scholar

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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

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D. Montgomery, Almost periodic transformation groups,, Trans. Amer. Math. Soc., 42 (1937), 322.  doi: 10.1090/S0002-9947-1937-1501924-0.  Google Scholar

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V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations,, Princeton University Press, (1960).   Google Scholar

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A. Weil, L'Integration Dans Les Groupes Topologiques et Ses Applications,, Actualitiés scientifiques, (1938).   Google Scholar

show all references

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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