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 alpha on a torus T-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 n epsilon Z(r) such that alpha(n) expands this direction. In this paper, we investigate what happens when this condition is removed; more generally, we consider a partial orbit {alpha(n) . x : n epsilon Omega} where Omega 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 r >= 3.