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Generic extensions of ergodic actionsстатья
- Автор: Ryzhikov V.V.
- Журнал: arXive e-print
- Номер: arXiv:2209.09160
- Год издания: 2022
- Аннотация: The article considers generic extensions of measure-preserving actions, particularly, the lifting of some action invariants. The work is encouraged by recent results of Austin, Glasner, Thouvenot, and Weiss. We prove that the P-entropy of the generic extensions with finite P-entropy, which is an invariant of the Kushnirenko entropy type, is infinite. A different approach is exploited to obtain the result of the above-mentioned authors that the generic extension of an action with zero classical entropy is not isomorphic to this action. It is shown that typical extensions preserve the singularity of spectrum, as well as some approximation and asymptotic properties of the base. We prove that the possibility to lift via the typical extensions the algebraic property "to be a composition of two involutions" depends on the statistical properties of the base. We consider also typical measurable automorphism families of the probability space (in short, communities) that arise by extending the identity operator. It is shown that for the typical community, under iterations, the automorphisms included in it behave coherently on some time sequences, although on other sequences they exhibit individual behavior. The article also formulates some questions, mentions the Parreau factor in connection both with the independent factor conjecture and the conjecture on generic lifting of the mixing property.
- Добавил в систему: Рыжиков Валерий Валентинович