Isomorphism rigidity of irreducible algebraic ℤd-actions
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- Kitchens, B. & Schmidt, K. Invent. math. (2000) 142: 559. doi:10.1007/PL00005793
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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 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.  and ). By combining these two theorems one obtains isomorphism rigidity of all irreducible, expansive and mixing algebraic ℤd-actions with d≥2.