## 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 1 2012 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics