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.

show all references

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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