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Generic properties of homeomorphisms preserving a given dynamical simplex

Part of: Topological dynamics

Published online by Cambridge University Press:  25 October 2021

JULIEN MELLERAY
JULIEN MELLERAY*
Affiliation:
Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, F-69622 Villeurbanne, France
*
e-mail: melleray@math.univ-lyon1.fr
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Abstract

Given a dynamical simplex K on a Cantor space X, we consider the set $G_K^*$ of all homeomorphisms of X which preserve all elements of K and have no non-trivial clopen invariant subset. Generalizing a theorem of Yingst, we prove that for a generic element g of $G_K^*$ the set of invariant measures of g is equal to K. We also investigate when there exists a generic conjugacy class in $G_K^*$ and prove that this happens exactly when K has only one element, which is the unique invariant measure associated to some odometer; and that in that case the conjugacy class of this odometer is generic in $G_K^*$ .

Keywords

topological dynamicsinvariant measuresminimal homeomorphismsBaire category

MSC classification

Primary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Secondary: 37B99: None of the above, but in this section
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 2 , February 2023 , pp. 646 - 662
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

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