ISOMORPHISM RIGIDITY OF IRREDUCIBLEALGEBRAIC Zd - ACTIONSBRUCE KITCHENS AND KLAUS

An irreducible algebraic Z d-action on a compact abelian group X is a Z d-action by automorphisms of X such that every closed,-invariant subgroup Y (X is nite. 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 aane. 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.