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Rigidity properties of $\zd$-actions on tori and solenoids
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## Rigidity properties of $\zd$-actions on tori and solenoids

### Manfred Einsiedler and Elon Lindenstrauss

**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.

*Copyright 2003 American Mathematical Society*

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#### Article Info

- ERA Amer. Math. Soc.
**09** (2003), pp. 99-110
- Publisher Identifier: S 1079-6762(03)00117-3
- 2000
*Mathematics Subject Classification*. Primary 37A35; Secondary 37A45
*Key words and phrases*. Entropy, invariant measures, invariant $\sigma$-algebras, measurable factors, joinings, toral automorphisms, solenoid automorphism
- Received by editors July 12, 2003
- Posted on October 14, 2003
- Communicated by Klaus Schmidt
- Comments (When Available)

**Manfred Einsiedler**

Department of Mathematics, Box 354350, University of Washington, Seattle, WA 98195

*E-mail address:* `einsiedl@math.washington.edu`

**Elon Lindenstrauss**

Department of Mathematics, Stanford University, Stanford, CA 94305

*Current address:* Courant Institute of Mathematical Sciences, 251 Mercer St., New York, NY 10012

*E-mail address:* `elonbl@member.ams.org`

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.

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