Topological Self-joinings of Cartan Actions by Toral Automorphisms

TL;DR

This paper classifies invariant, topologically transitive subsets under certain higher-rank toral automorphism actions, showing they are homogeneous except for a constructed counterexample in rank 2.

Contribution

It establishes a classification of invariant sets for higher-rank Cartan actions on tori, revealing their homogeneous structure and providing a counterexample in the rank 2 case.

Findings

Invariant sets are either the whole torus, rational points, or translates of invariant subtori.

A counterexample exists for rank 2 actions.

Higher-rank actions exhibit homogeneous invariant structures.

Abstract

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