Isomorphism rigidity of irreducible algebraic Zd-actions

An irreducible algebraic Z(d)-action alpha on a compact abelian group X is a Z(d)-action by automorphisms of X such that every closed, alpha -invariant subgroup Y not subset of or equal to X is finite. We prove the following result: if d greater than or equal to 2, then every measurable conjugacy between irreducible and mixing algebraic Z(d)-actions on compact zero-dimensional abelian groups is affine. For irreducible, expansive and mixing algebraic Z(d)-actions on compact connected abelian groups the analogous statement follows essentially from a result by Katok and Spatzier on invariant measures of such actions (cf. [4] and [3]). By combining these two theorems one obtains isomorphism rigidity of all irreducible, expansive and mixing algebraic Z(d)-actions with d greater than or equal to 2.