Isomorphism rigidity of irreducible algebraic ℤd-actions

Abstract.

An irreducible algebraic ℤ ^d -actionα on a compact abelian group X is a ℤ^d-action by automorphisms of X such that every closed, α-invariant subgroup Y⊊X is finite. We prove the following result: if d≥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≥2.