Abstract
AbstractBerend [Multi-invariant sets on tori. Trans. Amer. Math. Soc.280(2) (1983), 509–532] obtained necessary and sufficient conditions on a ℤr-action α on a torus 𝕋d 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 ℤr-action there is an element n∈ℤr such that αn expands this direction. In this paper, we investigate what happens when this condition is removed; more generally, we consider a partial orbit {αn⋅x:n∈Ω} where Ω 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 ℤr-actions on tori for r≥3 .
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