A characterization of regularly almost periodic minimal flows

Abstract. Regular almost periodicity in compact minimal abelian flows was characterized for the case of discrete acting group by W. Gottschalk and G. Hedlund and for the case of 0-dimensional phase space by W. Gottschalk a few decades ago. In 1995 J. Egawa gave characterizations for the case when the acting group is R. We extend Egawa's results to the case of an arbitrary abelian acting group and a not necessarily metrizable phase space. We then show how our statements imply previously known characterizations in each of the three special cases and give various other applications (characterization of regularly almost periodic functions on arbitrary abelian topological groups, classification of uniformly regularly almost periodic compact minimal Z-and R-flows, conditions equivalent with uniform regular almost periodicity, etc.).

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“…Thus in the case T = R we get the situation considered by Egawa in [9]. The following lemma is easy to prove.…”

Section: Theorem 710 For a Function F ∈ Ap (T ) The Following Are Ementioning

confidence: 75%

Characterizations of regular almost periodicity in compact minimal abelian flows

Miller

Rosenblatt

2004

Trans. Amer. Math. Soc.

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“…It should be noted here that the ε-error period set P(ε) in Def. 1(b) is not required to be a sub-semigroup of G. Otherwise it is named "uniform regular almost periodicity" and the latter is systematically studied in [4,9] when G is assumed to be a topological group.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of Bohr almost periodic motions of topological abelian semigroups

Dai

2016

Semigroup Forum

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“…Clearly this recurrence depends upon the topology of G and the strongest type of uniform recurrence occurs when G is provided with the discrete topology. It also should be noted here that if the syndetic set in Definition 1.1 is required to be a subgroup of G where G is assumed a topological group, then this kind of recurrence is named "regular almost periodicity" and the latter is systematically studied in [1,6].…”

mentioning

confidence: 99%

“…It should be noted that in some literature like [3,4], [5,Definition 3.38 and Remark 4.02- (1)], a uniformly recurrent point is sometimes called an almost periodic point (in the sense of von Neumann) for G X. However, this is much more weaker than the following one corresponding to the classical almost periodic function of Bohr: Definition 1.2.…”

mentioning

confidence: 99%

On uniformly recurrent motions of topological semigroup actions

Chen¹,

Dai²

2015

DCDS-A

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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: • For any neighborhood U of x, the return times set {g ∈ G : gx ∈ U } is syndetic of Furstenburg in G. • Given any ε > 0, there exists a finite subset K of G such that for each g in G, the ε-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 = R or Z. In addition, a counterexample is constructed to this statement, while X is merely a complete but not locally compact metric space.

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A characterization of regularly almost periodic minimal flows

Cited by 5 publications

References 5 publications

Characterizations of regular almost periodicity in compact minimal abelian flows

Characterizations of regular almost periodicity in compact minimal abelian flows

Dynamics of Bohr almost periodic motions of topological abelian semigroups

On uniformly recurrent motions of topological semigroup actions

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