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 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 Z^d -actions.
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Einsiedler et al. (2003) studied this question.