Rigidity of commutative non-hyperbolic actions by toral automorphisms

Berend [Multi-invariant sets on tori. Trans. Amer. Math. Soc. 280(2) (1983), 509-532] obtained necessary and sufficient conditions on a ℤ ^r-action on a torus ^d by toral automorphisms in order for every orbit to be either finite or dense. One of these conditions is that for every common eigendirection of the ℤ ^r-action there is an element ℤ ^r such that ⁿ expands this direction. In this paper, we investigate what happens when this condition is removed; more generally, we consider a partial orbit { ⁿ} where is a set of elements which acts in an approximately isometric way on a given set of eigendirections. This analysis is used in an essential way in the work of the author with E. Lindenstrauss [Topological self-joinings of Cartan actions by toral automorphisms. Preprint, 2010] classifying topological self-joinings of maximal ℤ ^r-actions on tori for rA3.