Outdated Archival Version

These pages are not updated anymore. They reflect the state of 20 August 2005. For the current production of this journal, please refer to http://www.math.psu.edu/era/.

This journal is archived by the American Mathematical Society. The master copy is available at http://www.ams.org/era/

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

TeX source (49.8K)
DVI (64.5K)
PostScript (161.8K)

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.