Rigidity of measurable structure for Zd-actions by automorphisms of a torus

Anatole Katok, Svetlana Katok, Klaus Schmidt

Research output: Chapter in Book/Report/Conference proceedingChapter


We show that for certain classes of actions of Zd, d ≥ 2, by automorphisms of the torus any measurable conjugacy has to be affine, hence measurable conjugacy implies algebraic conjugacy; similarly any measurable factor is algebraic, and algebraic and affine centralizers provide invariants of measurable conjugacy. Using the algebraic machinery of dual modules and information about class numbers of algebraic number fields we construct various examples of Zdactions by Bernoulli automorphisms whose measurable orbit structure is rigid, including actions which are weakly isomorphic but not isomorphic. We show that the structure of the centralizer for these actions may or may not serve as a distinguishing measure-theoretic invariant.

Original languageEnglish (US)
Title of host publicationThe Collected Works of Anatole Katok
Subtitle of host publicationIn 2 Volumes
PublisherWorld Scientific Publishing Co.
Number of pages28
ISBN (Electronic)9789811238079
ISBN (Print)9789811238062
StatePublished - Jan 1 2024

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • General Engineering

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