Generic extensions of ergodic systems

V. V. Ryzhikov

Sbornik. Mathematics, Tome 214 (2023) no. 10, pp. 1442-1457

Résumé

The paper is devoted to problems concerning the generic properties of extensions of dynamical systems with invariant measures. It is proved that generic extensions preserve the singularity of the spectrum, the mixing property and some other asymptotic properties. It is discovered that the preservation of algebraic properties generally depends on statistical properties of the base. It is established that the $P$-entropy of a generic extension is infinite. This fact yields a new proof of the result due to Weiss, Glasner, Austin and Thouvenot on the nondominance of deterministic actions. Generic measurable families of automorphisms of a probability space are considered. It is shown that the asymptotic behaviour of representatives of a generic family is characterized by a combination of dynamic conformism and dynamic individualism. Bibliography: 15 titles.

Détail

Keywords: ergodic action, $P$-entropy, mixing, spectrum, generic properties of extensions.

Bibliographie

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