Isomorphism rigidity of commuting automorphisms

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.

14 pages, no figures