Isomorphism rigidity of commuting automorphisms

Bhattacharya, Siddhartha

doi:10.1090/S0002-9947-08-04597-2

Isomorphism rigidity of commuting automorphisms
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Trans. Amer. Math. Soc. 360 (2008), 6319-6329 Request permission

Abstract:

Let $d > 1$, and let $(X,\alpha )$ and $(Y,\beta )$ 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 $\alpha$ and $\beta$ 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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