Topological Self-joinings of Cartan Actions by Toral Automorphisms

We show that if r ≥ 3 and α is a faithful Z-Cartan action on a torus T by automorphisms, then any closed subset of (T) which is invariant and topologically transitive under the diagonal Z-action by α is homogeneous, in the sense that it is either the full torus (T), 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.