Rigidity of measurable structure for \mathbb Z^d-actions by automorphisms of a torus |
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| Authors: | A Katok S Katok K Schmidt |
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| Institution: | (1) The Pennsylvania State University, Department of Mathematics, University Park, PA 16802, USA katok_a@math.psu.edu , US;(2) The Pennsylvania State University, Department of Mathematics, University Park, PA 16802, USA katok_s@math.psu.edu , US;(3) University of Vienna, Mathematics Institute, Strudlhofgasse 4, A-1090 Vienna, Austria klaus.schmidt@univie.ac.at , AT |
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| Abstract: | We show that for certain classes of actions of , 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 -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.
Received: March 12, 2002 |
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| Keywords: | , Commuting hyperbolic toral automorphisms, isomorphism rigidity of $\mathbb Z^d$-actions, |
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