Rigidity of Commutative Non-hyperbolic Actions by Toral Automorphisms

TL;DR

This paper explores the rigidity properties of certain toral automorphism actions when a key expansion condition is removed, extending understanding of orbit structures and contributing to classification results in higher-rank dynamics.

Contribution

It generalizes previous conditions for orbit density and finiteness by analyzing partial orbits under approximate identity actions, aiding classification of self-joinings.

Findings

Partial orbit analysis reveals new rigidity phenomena.

Removal of expansion condition affects orbit structure classification.

Supports classification of higher-rank toral actions with Lindenstrauss.

Abstract

Berend gives necessary and sufficient conditions on a $Z^r$-action $\alpha$ on a torus $T^d$ by toral automorphisms in order for every orbit be either finite or dense. One of these conditions is that on every common eigendirection of the $Z^r$-action there is an element $\mathbf n \in Z^r$ so that $\alpha^\mathbf n$ expands this direction. In this paper, we investigate what happens when this condition is removed; more generally, we consider a partial orbit $\{\alpha^\mathbf n.x : \mathbf n \in \Omega\}$ where $\Omega$ is a set of elements which acts approximately as the identity on a given set of eigendirections. This analysis is used in an essential way in the work of the author with E. Lindenstrauss classifying topological self-joinings of maximal $Z^r$-actions on tori for $r\geq 3$.