## Abstract

We show that for certain classes of actions of Z^{d}, 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 language | English (US) |
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Title of host publication | The Collected Works of Anatole Katok |

Subtitle of host publication | In 2 Volumes |

Publisher | World Scientific Publishing Co. |

Pages | 2153-2180 |

Number of pages | 28 |

Volume | 2 |

ISBN (Electronic) | 9789811238079 |

ISBN (Print) | 9789811238062 |

State | Published - Jan 1 2024 |

## All Science Journal Classification (ASJC) codes

- General Mathematics
- General Engineering