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Isomorphism rigidity of commuting automorphisms

Siddhartha Bhattacharya
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Abstract

Let $d > 1$, and let $(X,α)$ and $(Y,β)$ be two zero-entropy ${\mathbb{Z}}^d$-actions on compact abelian groups by $d$ commuting automorphisms. We show that if all lower rank subactions of $α$ and $β$ have completely positive entropy, then any measurable equivariant map from $X$ to $Y$ is an affine map. In particular, two such actions are measurably conjugate if and only if they are algebraically conjugate.

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Siddhartha Bhattacharya. 2004-12-01. Isomorphism rigidity of commuting automorphisms. https://arxiv.org/abs/math/0412026

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