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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 Z dactions 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.

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We study a two-parameter family of one-dimensional maps and related (a, b)-continued fractions suggested for consideration by Don Zagier. We prove that the associated natural extension maps have attractors with finite rectangular structure for the entire parameter set except for a Cantor-like set of one-dimensional Lebesgue zero measure that we completely describe. We show that the structure of these attractors can be "computed" from the data (a, b), and that for a dense open set of parameters the Reduction theory conjecture holds, i.e., every point is mapped to the attractor after finitely many iterations. We also show how this theory can be applied to the study of invariant measures and ergodic properties of the associated Gauss-like maps. CONTENTS 1. Introduction 637 2. Theory of (a, b)-continued fractions 640 3. Attractor set for F a,b 644 4. Cycle property 647 5. Finiteness condition implies finite rectangular structure 655 6. Finite rectangular structure of the attracting set 666 7. Reduction theory conjecture 670 8. Set of exceptions to the finiteness condition 671 9. Invariant measures and ergodic properties 687 References 689