Abstract:
We answer positively a question of Ryzhikov, namely we show that being a relatively weakly mixing extension is a comeager property in the Polish group of measure preserving transformations. We study some related classes of ergodic transformations and their interrelations. In the second part of the paper we show that for a fixed ergodic T with property A, a generic extension ˆT of T also has the property A. Here A stands for each of the following properties: (i) having the same entropy as T, (ii) Bernoulli, (iii) K, and (iv) loosely Bernoulli. References: 46 entries.
Key words and phrases:
relative weak mixing, comeager properties, prime dynamical systems, Bernoulli systems, K-systems, loosely Bernoulli systems.
\Bibitem{GlaThoWei21}
\by E.~Glasner, J.-P.~Thouvenot, B.~Weiss
\paper On some generic classes of ergodic measure preserving transformations
\serial Tr. Mosk. Mat. Obs.
\yr 2021
\vol 82
\issue 1
\pages 19--44
\publ MCCME
\publaddr M.
\mathnet{http://mi.mathnet.ru/mmo645}
\transl
\jour Trans. Moscow Math. Soc.
\yr 2021
\vol 82
\pages 15--36
\crossref{https://doi.org/10.1090/mosc/312}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85127504467}
Linking options:
https://www.mathnet.ru/eng/mmo645
https://www.mathnet.ru/eng/mmo/v82/i1/p19
This publication is cited in the following 6 articles:
ADAM LOTT, “Zero entropy actions of amenable groups are not dominant”, Ergod. Th. Dynam. Sys., 44:2 (2024), 630
V. V. Ryzhikov, “Self-joinings and generic extensions of ergodic systems”, Funct. Anal. Appl., 57:3 (2023), 236–247
V. V. Ryzhikov, “Generic extensions of ergodic systems”, Sb. Math., 214:10 (2023), 1442–1457
TIM AUSTIN, ELI GLASNER, JEAN-PAUL THOUVENOT, BENJAMIN WEISS, “An ergodic system is dominant exactly when it has positive entropy”, Ergod. Th. Dynam. Sys., 43:10 (2023), 3216
V. V. Ryzhikov, “Tensor simple spectrum of unitary flows”, Funct. Anal. Appl., 56:4 (2022), 327–330
V. V. Ryzhikov, “Compact families and typical entropy invariants of measure-preserving actions”, Trans. Moscow Math. Soc., 82 (2021), 117–123