Abstract
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.
Original language | English (US) |
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Pages (from-to) | 1752-1782 |
Number of pages | 31 |
Journal | Ergodic Theory and Dynamical Systems |
Volume | 32 |
Issue number | 5 |
DOIs | |
State | Published - Oct 2012 |
All Science Journal Classification (ASJC) codes
- General Mathematics
- Applied Mathematics