Isomorphism rigidity of irreducible algebraic ℤ^d-actions

Abstract

An irreducible algebraic ℤd -action α on a compact abelian group X is a ℤd-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 ℤd-actions on compact zero-dimensional abelian groups is affine. For irreducible, expansive and mixing algebraic ℤd-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 ℤd-actions with d ≥ 2.