Abstract.
An irreducible algebraic ℤ d -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. [4] and [3]). By combining these two theorems one obtains isomorphism rigidity of all irreducible, expansive and mixing algebraic ℤd-actions with d≥2.
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Oblatum 30-IX-1999 & 4-V-2000¶Published online: 16 August 2000
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Kitchens, B., Schmidt, K. Isomorphism rigidity of irreducible algebraic ℤd-actions. Invent. math. 142, 559–577 (2000). https://doi.org/10.1007/PL00005793
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DOI: https://doi.org/10.1007/PL00005793