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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 |
$\bullet$ 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.
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. |
[2] |
H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory,, Princeton University Press, (1981).
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[3] |
W. H. Gottschalk, Almost periodic points with respect to transformation semi-groups,, Annals of Math., 47 (1946), 762.
doi: 10.2307/1969233. |
[4] |
W. H. Gottschalk, A survey of minimal sets,, Ann. Inst. Fourier, 14 (1964), 53.
doi: 10.5802/aif.160. |
[5] |
W. H. Gottschalk and G. A. Hedlund, Topological Dynamics,, Amer. Math. Soc. Coll. Publ., (1955).
|
[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. |
[7] |
D. Montgomery, Almost periodic transformation groups,, Trans. Amer. Math. Soc., 42 (1937), 322.
doi: 10.1090/S0002-9947-1937-1501924-0. |
[8] |
V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations,, Princeton University Press, (1960).
|
[9] |
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. |
[2] |
H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory,, Princeton University Press, (1981).
|
[3] |
W. H. Gottschalk, Almost periodic points with respect to transformation semi-groups,, Annals of Math., 47 (1946), 762.
doi: 10.2307/1969233. |
[4] |
W. H. Gottschalk, A survey of minimal sets,, Ann. Inst. Fourier, 14 (1964), 53.
doi: 10.5802/aif.160. |
[5] |
W. H. Gottschalk and G. A. Hedlund, Topological Dynamics,, Amer. Math. Soc. Coll. Publ., (1955).
|
[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. |
[7] |
D. Montgomery, Almost periodic transformation groups,, Trans. Amer. Math. Soc., 42 (1937), 322.
doi: 10.1090/S0002-9947-1937-1501924-0. |
[8] |
V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations,, Princeton University Press, (1960).
|
[9] |
A. Weil, L'Integration Dans Les Groupes Topologiques et Ses Applications,, Actualitiés scientifiques, (1938). Google Scholar |
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