Topological self-joinings of Cartan actions by toral automorphisms

Abstract

We show that if $r\ge 3$ and $\alpha$ is a faithful ${\mathbb{Z}}^r$-Cartan action on a torus ${\mathbb{T}}^d$ by automorphisms, then any closed subset of $({\mathbb{T}}^d{)}^2$ which is invariant and topologically transitive under the diagonal ${\mathbb{Z}}^r$-action by $\alpha$ is homogeneous, in the sense that it is either the full torus $({\mathbb{T}}^d{)}^2$, or a finite set of rational points, or a finite disjoint union of parallel translates of some $d$-dimensional invariant subtorus. A counterexample is constructed for the rank $2$ case.