2000/03/05 by Anatole Katok, Svetlana Katok, Katok, Anatole +3 · 1 citation
Mathematics · Physics and Astronomy · #11R29 (Secondary) #37C85 (Primary) 22F10 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Quantum chaos and dynamical systems #math.DS #math.NT #msc:11R29 #msc:22F10 #msc:37C85
paper · pdf · doi:10.48550/arxiv.math/0003032
30 pages, AMS-LaTeX
arxiv created 2000/03/05 · openalex publication_date 2000/03/05 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 consruct various examples of Zd-actions 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.