Almost periodic points and minimal sets in topological spaces

Chikara Fujita

Japan

University of Tsukuba

Institute of Mathematics, University of Tsukuba, Ibaraki, 305-8571 Japan.

Hisao Kato

Japan

University of Tsukuba

Institute of Mathematics, University of Tsukuba, Ibaraki, 305-8571 Japan.

Submitted: 2013-09-30

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Accepted: 2013-09-30

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DOI: https://doi.org/10.4995/agt.2009.1737
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Keywords:

Almost periodic points, Minimal sets, w-regular space.

Supporting agencies:

This research was not funded

Abstract:

In the paper [Almost periodic points and minimal sets in w-regular spaces, Topology Appl. 154 (2007), 2873–2879], Mai and Sun showed that several known results concerning almost periodic points and minimal sets of maps can be generalized from regular spaces to w-regular spaces. Also, they have three unsolved problems. In this paper, we answer to all problems which remain unsolved in the paper of Mai and Sun. In fact we prove some general theorems which give counte rexamples of the problems.
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References:

G. D. Birkhoff, Dynamical Systems, revised editions, Amer. Math. Soc. Colloq. Publ., vol. IX, Amer. Math. Soc., Providence, RI, 1966.

W. H. Gottschalk, Orbit-closure decompositions and almost periodic properties, Bull. Amer. Math. Soc. 50 (1944), 915–919. http://dx.doi.org/10.1090/S0002-9904-1944-08262-1

W. H. Gottschalk, Almost periodic points with respect to transformation semi-groups, Ann. Math. 47 (1946), 762–766. http://dx.doi.org/10.2307/1969233

J. L. Kelley, General Topology, Springer-Verlag, New York, 1975.

J.-H. Mai and W.-H. Sun, Almost periodic points and minimal sets in !-regular spaces, Topology Appl. 154 (2007), 2873–2879. http://dx.doi.org/10.1016/j.topol.2007.06.007

A. Fedeli and A. Le Donne, On almost periodic orbits and minimal sets, Topology Appl. 156 (2008), no. 2, 473–475.

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