Abstract.
We consider mixing ℤd-actions on compact zero-dimensional abelian groups by automorphisms. Rigidity of invariant measures does not hold for such actions in general; we present conditions which force an invariant measure to be Haar measure on an affine subset. This is applied to isomorphism rigidity for such actions. We develop a theory of halfspace entropies which plays a similar role in the proof to that played by invariant foliations in the proof of rigidity for smooth actions.
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Einsiedler, M. Isomorphism and Measure Rigidity for Algebraic Actions on Zero-Dimensional Groups. Monatsh. Math. 144, 39–69 (2005). https://doi.org/10.1007/s00605-004-0248-1
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DOI: https://doi.org/10.1007/s00605-004-0248-1