Isomorphism rigidity of commuting automorphisms

Let d > 1, and let (X, alpha) and (Y, beta) be two zero-entropy 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.