Treffer: Isomorphism rigidity of irreducible algebraic \(\mathbb{Z}^d\)-actions

An irreducible algebraic \(\mathbb{Z}^d\)-action on a compact abelian group \(X\) is a \(\mathbb{Z}^d\)-action by automorphism of \(X\) such that every closed \(\alpha\)-invariant subgroup of \(X\) is finite. An algebraic \(\mathbb{Z}^d\)-action is expansive if there is an open set \({\mathcal O}\subset X\) such that \(\cap_{n\in \mathbb{Z}^d} \alpha^{-n} ({\mathcal O})\) consists of the identity element of \(X\). The action \(\alpha\) is said to be mixing (ergodic) if the Haar measure of \(X\) is mixing (ergodic) under the action of \(\alpha\). In order to classify expansive algebraic \(\mathbb{Z}^d\)-actions the authors introduced the notion of measurable conjugacy of such actions. The authors prove that every measurable conjugacy between irreducible mixing \(\mathbb{Z}^d\)-actions on compact zero-dimensional abelian groups is affine provided \(d\geq 2\). Combining this result with previous work of Katok and Spatzier the authors obtain isomorphism rigidity of all irreducible expansive and mixing \(\mathbb{Z}^d\)-actions when \(d\geq 2\).