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Complete regularity of Ellis semigroups of $\mathbb{Z} $-actions

Part of: Semigroups Connections with other structures, applications Topological dynamics

Published online by Cambridge University Press:  13 November 2020

MARCY BARGE  and
JOHANNES KELLENDONK Open the ORCID record for JOHANNES KELLENDONK [Opens in a new window]
MARCY BARGE
Affiliation:
Montana State University, Department of Mathematical Sciences, Bozeman, MT59717, USA (e-mail: barge@math.montana.edu)
JOHANNES KELLENDONK*
Affiliation:
Univerisité de Lyon, Université Claude Bernard Lyon 1, Institute Camille Jordan, CNRS UMR 5208, 69622Villeurbanne, France
*
e-mail: kellendonk@math.univ-lyon1.fr
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Abstract

It is shown that the Ellis semigroup of a $\mathbb Z$ -action on a compact totally disconnected space is completely regular if and only if forward proximality coincides with forward asymptoticity and backward proximality coincides with backward asymptoticity. Furthermore, the Ellis semigroup of a $\mathbb Z$ - or $\mathbb R$ -action for which forward proximality and backward proximality are transitive relations is shown to have at most two left minimal ideals. Finally, the notion of near simplicity of the Ellis semigroup is introduced and related to the above.

Keywords

enveloping semigroupcompletely regular semigrouptopological dynamics

MSC classification

Primary: 54H20: Topological dynamics
Secondary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.) 20M17: Regular semigroups
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 12 , December 2021 , pp. 3593 - 3609
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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