Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2023
|
| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2312.13165 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866914958705426432 |
|---|---|
| author | Tumarkin, Yuriy |
| author_facet | Tumarkin, Yuriy |
| contents | We consider a special case of the question of classification of invariant Radon measures of $\mathbb{Z}^m$-valued skew-products over interval exchange transformations, which arise as Poincaré sections of the linear flow on periodic infinite translation surfaces. In the case of periodic type skew-products, we obtain a full classification of ergodic invariant Radon measures, showing them to be precisely the Maharam measures, a family of measures parametrised by $\mathbb{R}^m$. For the proof we translate Rauzy-Veech renormalisation for skew-products into the symbolic language of the adic coding, and apply a symbolic result of Aaronson, Nakada, Sarig and Solomyak. Further, we use this language and a new extension of the Rauzy-Veech cocycle to find an explicit form for the Maharam measures and deduce the weak*-continuity of the measures depending on the parameter. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_13165 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Ergodic measures for periodic type $\mathbb{Z}^m$-skew-products over Interval Exchange Transformations Tumarkin, Yuriy Dynamical Systems We consider a special case of the question of classification of invariant Radon measures of $\mathbb{Z}^m$-valued skew-products over interval exchange transformations, which arise as Poincaré sections of the linear flow on periodic infinite translation surfaces. In the case of periodic type skew-products, we obtain a full classification of ergodic invariant Radon measures, showing them to be precisely the Maharam measures, a family of measures parametrised by $\mathbb{R}^m$. For the proof we translate Rauzy-Veech renormalisation for skew-products into the symbolic language of the adic coding, and apply a symbolic result of Aaronson, Nakada, Sarig and Solomyak. Further, we use this language and a new extension of the Rauzy-Veech cocycle to find an explicit form for the Maharam measures and deduce the weak*-continuity of the measures depending on the parameter. |
| title | Ergodic measures for periodic type $\mathbb{Z}^m$-skew-products over Interval Exchange Transformations |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2312.13165 |