### 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) |
---|---|

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 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

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*Ergodic Theory and Dynamical Systems*, vol. 32, no. 5, pp. 1752-1782. https://doi.org/10.1017/S0143385711000484

**Rigidity of commutative non-hyperbolic actions by toral automorphisms.** / Wang, Zhiren.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Rigidity of commutative non-hyperbolic actions by toral automorphisms

AU - Wang, Zhiren

PY - 2012/10/1

Y1 - 2012/10/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84866038085&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84866038085&partnerID=8YFLogxK

U2 - 10.1017/S0143385711000484

DO - 10.1017/S0143385711000484

M3 - Article

AN - SCOPUS:84866038085

VL - 32

SP - 1752

EP - 1782

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 5

ER -