Affine random walks on the torus
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2020
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909826168127488 |
|---|---|
| author | He, Weikun Lakrec, Tsviqa Lindenstrauss, Elon |
| author_facet | He, Weikun Lakrec, Tsviqa Lindenstrauss, Elon |
| contents | We consider random walks on the torus arising from the action of the group of affine transformations. We give a quantitative equidistribution result for this random walk under the assumption that the Zariski closure of the group generated by the linear part acts strongly irreducibly on $\mathbb{R}^d$ and is either Zariski connected or contains a proximal element. Specifically, we give quantitative estimates (depending only on the linear part of the random walk) for how fast the random walk equidistributes unless the initial point and the translation part of the affine transformations can be perturbed so that the random walk is trapped in a finite orbit of small cardinality. In particular, we prove that the random walk equidistributes in law to the Haar measure if and only if the random walk is not trapped in a finite orbit. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2003_03743 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Affine random walks on the torus He, Weikun Lakrec, Tsviqa Lindenstrauss, Elon Dynamical Systems We consider random walks on the torus arising from the action of the group of affine transformations. We give a quantitative equidistribution result for this random walk under the assumption that the Zariski closure of the group generated by the linear part acts strongly irreducibly on $\mathbb{R}^d$ and is either Zariski connected or contains a proximal element. Specifically, we give quantitative estimates (depending only on the linear part of the random walk) for how fast the random walk equidistributes unless the initial point and the translation part of the affine transformations can be perturbed so that the random walk is trapped in a finite orbit of small cardinality. In particular, we prove that the random walk equidistributes in law to the Haar measure if and only if the random walk is not trapped in a finite orbit. |
| title | Affine random walks on the torus |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2003.03743 |