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Maximally highly proximal flows
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 41 / Issue 7 / July 2021
- Published online by Cambridge University Press:
- 18 May 2020, pp. 2220-2240
- Print publication:
- July 2021
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- Article
- Export citation
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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.
