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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Characterizations of regular almost periodicity in compact minimal abelian flows


Authors: Alica Miller and Joseph Rosenblatt
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 4909-4929
MSC (2000): Primary 37B05, 43A60; Secondary 43A40, 54H20
Published electronically: February 4, 2004
MathSciNet review: 2084405
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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 $\mathbb{R}$. 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 $\mathbb{Z}$- and $\mathbb{R}$-flows, conditions equivalent with uniform regular almost periodicity, etc.).


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Additional Information

Alica Miller
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
Email: amiller@math.uiuc.edu

Joseph Rosenblatt
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
Email: jrsnbltt@math.uiuc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03538-X
PII: S 0002-9947(04)03538-X
Keywords: Compact minimal flows, almost periodic, regularly almost periodic, almost automorphic, almost $1-1$, eigenvalues, $0$-dimensional, Bohr compactification
Received by editor(s): November 3, 2002
Received by editor(s) in revised form: June 19, 2003
Published electronically: February 4, 2004
Article copyright: © Copyright 2004 American Mathematical Society