Characterizations of regular almost periodicity in compact minimal abelian flows

Abstract

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.).

Dynamics of Bohr almost periodic motions of topological abelian semigroups

“…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.…”

On uniformly recurrent motions of topological semigroup actions

“…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].…”