Inventiones mathematicae

, Volume 142, Issue 3, pp 559–577

Isomorphism rigidity of irreducible algebraic ℤd-actions

  • Bruce Kitchens
  • Klaus Schmidt

DOI: 10.1007/PL00005793

Cite this article as:
Kitchens, B. & Schmidt, K. Invent. math. (2000) 142: 559. doi:10.1007/PL00005793


An irreducible algebraic ℤd-actionα on a compact abelian group X is a ℤd-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 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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Bruce Kitchens
    • 1
  • Klaus Schmidt
    • 2
  1. 1.Mathematical Sciences Department, IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, USA (e-mail:
  2. 2.Mathematics Institute, University of Vienna, Strudlhofgasse 4, A-1090 Vienna, AustriaAT
  3. 3.Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, A-1090 Vienna, Austria (e-mail: