Topological self-joinings of cartan actions by toral automorphisms

We show that if r ≥ 3 and α is a faithful ℤ( ^r-Cartan action on a torus T{double-struck} ^d by automorphisms, then any closed subset of (T{double-struck} ^d) ² which is invariant and topologically transitive under the diagonal ℤ ^r -action by α is homogeneous, in the sense that it is either the full torus (T{double-struck} ^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.