We show that if and is a faithful -Cartan action on a torus by automorphisms, then any closed subset of which is invariant and topologically transitive under the diagonal -action by is homogeneous, in the sense that it is either the full torus , or a finite set of rational points, or a finite disjoint union of parallel translates of some -dimensional invariant subtorus. A counterexample is constructed for the rank case.
"Topological self-joinings of Cartan actions by toral automorphisms." Duke Math. J. 161 (7) 1305 - 1350, 15 May 2012. https://doi.org/10.1215/00127094-1593290