Joinings of the action of the group \(GL(n,\mathbb{Z})\) on the \(n\)-dimensional torus
Let \(\Psi=\{\Psi^{t}\}\) be a group action of \(GL(n,Z)\) on the \(n\)-dimensional torus \(\mathbb T^{n}\) by the Haar measure \(\lambda\) preserving transformations: \((\Psi^{t}\xi)^{j}:=\sum_{k=1}^{n}t_{k}^{j}\xi^{k}\), where \(\xi\in\mathbb T^{n}\), and \(t\) be an element of the group \(GL(n,Z)\). A joining of order \(m\) of the action \(\Psi\) is a Borel measure \(\nu\) on \((\mathbb T^{n})^{m}\) which is \(\Psi\)-invariant and has all one-dimensional projections equal to \(\lambda\). In [\textit{K. K. Park}, Proc. Am. Math. Soc 114, No. 4, 955-963 (1992; Zbl 0749.28009)] it was shown that any ergodic joining of the \(GL(2,Z)\) action different from a direct product of a Haar measure on \(\mathbb T^{m}\) by itself, is concentrated on some proper closed subgroup of type \(H\times H\). The main aim of the present paper is to generalize this result for the case \(n\geq 2\) and to give a classification of ergodic joinings.
scientific article; zbMATH DE number 1228285