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

Abstract

We show that Haar measure is a unique measure on a torus or more generally a solenoid X X invariant under a not virtually cyclic totally irreducible Z d \mathbb Z^d -action by automorphisms of X 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 Z d \mathbb Z^d -actions.