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Maximally highly proximal flows

Part of: Topological dynamics Set theory Connections with other structures, applications

Published online by Cambridge University Press:  18 May 2020

ANDY ZUCKER Open the ORCID record for ANDY ZUCKER [Opens in a new window]
ANDY ZUCKER*
Affiliation:
Université Paris Diderot, IMJ-PRG, 8 place Aurélie Nemours, Bat. Sophie Germain, Paris, 75205, France email andrew.zucker@imj-prg.fr
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Abstract

For $G$ a Polish group, we consider $G$-flows which either contain a comeager orbit or have all orbits meager. We single out a class of flows, the maximally highly proximal (MHP) flows, for which this analysis is particularly nice. In the former case, we provide a complete structure theorem for flows containing comeager orbits, generalizing theorems of Melleray, Nguyen Van Thé, and Tsankov and of Ben Yaacov, Melleray, and Tsankov. In the latter, we show that any minimal MHP flow with all orbits meager has a metrizable factor with all orbits meager, thus ‘reflecting’ complicated dynamical behavior to metrizable flows. We then apply this to obtain a structure theorem for Polish groups whose universal minimal flow is distal.

Keywords

topological dynamicscomeager orbitshighly proximal extensions

MSC classification

Primary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Secondary: 54H20: Topological dynamics 03E15: Descriptive set theory
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 7 , July 2021 , pp. 2220 - 2240
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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