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Minimality for actions of abelian semigroups on compact spaces with a free interval

Published online by Cambridge University Press:  06 March 2018

MATÚŠ DIRBÁK Open the ORCID record for MATÚŠ DIRBÁK [Opens in a new window] ,
ROMAN HRIC ,
PETER MALIČKÝ ,
L’UBOMÍR SNOHA  and
VLADIMÍR ŠPITALSKÝ
MATÚŠ DIRBÁK
Affiliation:
Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, 974 01 Banská Bystrica, Slovakia email Matus.Dirbak@umb.sk, Roman.Hric@umb.sk, Peter.Malicky@umb.sk, Lubomir.Snoha@umb.sk, Vladimir.Spitalsky@umb.sk
ROMAN HRIC
Affiliation:
Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, 974 01 Banská Bystrica, Slovakia email Matus.Dirbak@umb.sk, Roman.Hric@umb.sk, Peter.Malicky@umb.sk, Lubomir.Snoha@umb.sk, Vladimir.Spitalsky@umb.sk
PETER MALIČKÝ
Affiliation:
Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, 974 01 Banská Bystrica, Slovakia email Matus.Dirbak@umb.sk, Roman.Hric@umb.sk, Peter.Malicky@umb.sk, Lubomir.Snoha@umb.sk, Vladimir.Spitalsky@umb.sk
L’UBOMÍR SNOHA
Affiliation:
Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, 974 01 Banská Bystrica, Slovakia email Matus.Dirbak@umb.sk, Roman.Hric@umb.sk, Peter.Malicky@umb.sk, Lubomir.Snoha@umb.sk, Vladimir.Spitalsky@umb.sk
VLADIMÍR ŠPITALSKÝ
Affiliation:
Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, 974 01 Banská Bystrica, Slovakia email Matus.Dirbak@umb.sk, Roman.Hric@umb.sk, Peter.Malicky@umb.sk, Lubomir.Snoha@umb.sk, Vladimir.Spitalsky@umb.sk
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Abstract

We study minimality for continuous actions of abelian semigroups on compact Hausdorff spaces with a free interval. First, we give a necessary and sufficient condition for such a space to admit a minimal action of a given abelian semigroup. Further, for actions of abelian semigroups we provide a trichotomy for the topological structure of minimal sets intersecting a free interval.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 11 , November 2019 , pp. 2968 - 2982
Copyright
© Cambridge University Press, 2018 

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References

Beklaryan, L. A.. Groups of homeomorphisms of the line and the circle, topological characteristics and metric invariants. Russian Math. Surveys 59(4) (2004), 599–660.Google Scholar
Dirbák, M., Snoha, L’. and Špitalský, V.. Minimality, transitivity, mixing and topological entropy on spaces with a free interval. Ergod. Th. & Dynam. Sys. 33(6) (2013), 1786–1812.10.1017/S0143385712000442Google Scholar
Fuchs, L.. Infinite Abelian Groups. Vol. I (Pure and Applied Mathematics, 36) . Academic Press, Burlington, MA, 1970.Google Scholar
Ghys, É.. Groups acting on the circle. Enseign. Math. 47(3/4) (2001), 329–407.Google Scholar
Glasner, E.. Short proofs of theorems of Malyutin and Margulis. Proc. Amer. Math. Soc. 145(12) (2017), 5463–5467.Google Scholar
Kolyada, S., Snoha, L’. and Trofimchuk, S.. Noninvertible minimal maps. Fund. Math. 168(2) (2001), 141–163.10.4064/fm168-2-5Google Scholar
Malyutin, A.. Classification of the group actions on the real line and circle. St. Petersburg Math. J. 19(2) (2008), 279–296.10.1090/S1061-0022-08-00999-0Google Scholar
Margulis, G. A.. Free subgroups of the homeomorphism group of the circle. C. R. Acad. Sci. Paris Sér. I Math. 331(9) (2000), 669–674.Google Scholar
Navas, A.. Groups of Circle Diffeomorphisms (Chicago Lectures in Mathematics) . University of Chicago Press, Chicago, 2011.Google Scholar
Shinohara, K.. On the minimality of semigroup actions on the interval which are C 1 -close to the identity. Proc. Lond. Math. Soc. (3) 109(5) (2014), 1175–1202.Google Scholar