Inventiones mathematicae

, Volume 142, Issue 3, pp 559-577

First online:

Isomorphism rigidity of irreducible algebraic ℤd-actions

  • Bruce KitchensAffiliated withMathematical Sciences Department, IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, USA (e-mail:
  • , Klaus SchmidtAffiliated withMathematics Institute, University of Vienna, Strudlhofgasse 4, A-1090 Vienna, Austria

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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 YX 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.