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

Einsiedler, Manfred; Lindenstrauss, Elon

doi:10.1090/S1079-6762-03-00117-3

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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