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Rigidity properties of $\mathbb{Z} ^{d}$-actions on tori and solenoids


Authors: Manfred Einsiedler and Elon Lindenstrauss
Journal: Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 99-110
MSC (2000): Primary 37A35; Secondary 37A45
DOI: https://doi.org/10.1090/S1079-6762-03-00117-3
Published electronically: October 14, 2003
MathSciNet review: 2029471
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Abstract: We show that Haar measure is a unique measure on a torus or more generally a solenoid $X$ invariant under a not virtually cyclic totally irreducible $\mathbb Z^d$-action by automorphisms of $X$ such that at least one element of the action acts with positive entropy. We also give a corresponding theorem in the non-irreducible case. These results have applications regarding measurable factors and joinings of these algebraic $\mathbb Z^d$-actions.


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Additional Information

Manfred Einsiedler
Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, WA 98195
Email: einsiedl@math.washington.edu

Elon Lindenstrauss
Affiliation: Department of Mathematics, Stanford University, Stanford, CA 94305
Address at time of publication: Courant Institute of Mathematical Sciences, 251 Mercer St., New York, NY 10012
Email: elonbl@member.ams.org

DOI: https://doi.org/10.1090/S1079-6762-03-00117-3
Keywords: Entropy, invariant measures, invariant $\sigma$-algebras, measurable factors, joinings, toral automorphisms, solenoid automorphism
Received by editor(s): July 12, 2003
Published electronically: October 14, 2003
Additional Notes: E.L. is supported in part by NSF grant DMS-0140497. The two authors gratefully acknowledge the hospitality of Stanford University and the University of Washington, respectively
Communicated by: Klaus Schmidt
Article copyright: © Copyright 2003 American Mathematical Society

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