Rigidity of commutative non-hyperbolic actions by toral automorphisms

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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 languageEnglish (US)
Pages (from-to)1752-1782
Number of pages31
JournalErgodic Theory and Dynamical Systems
Volume32
Issue number5
DOIs
StatePublished - Oct 1 2012

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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