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

Published online by Cambridge University Press:  13 November 2020

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

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.

Type
Original Article
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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