Linearizations of periodic point free distal homeomorphisms on the annulus
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912134877675520 |
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| author | Shi, Enhui Xu, Hui Yu, Ziqi |
| author_facet | Shi, Enhui Xu, Hui Yu, Ziqi |
| contents | Let $\mathbb{A}$ be an annulus in the plane $\mathbb R^2$ and $g:\mathbb{A}\rightarrow \mathbb{A}$ be a boundary components preserving homeomorphism which is distal and has no periodic points. In \cite{SXY}, the authors show that there is a continuous decomposition $\mathcal P$ of $\mathbb{A}$ into $g$-invariant circles such that all the restrictions of $g$ on them share a common irrational rotation number (also called the rotation number of $g$) and all these circles are linearly ordered by the inclusion relation on the sets of bounded components of their complements in $\mathbb R^2$. In this note, we show that if the decomposition $\mathcal P$ above has a continuous section, then $g$ can be linearized, that is it is topologically conjugate to a rigid rotation on $\mathbb{A}$. For every irrational number $α\in (0, 1)$, we show the existence of such a distal homeomorphism $g$ on $\mathbb{A}$ that it cannot be linearized and its rotation number is $α$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_18360 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Linearizations of periodic point free distal homeomorphisms on the annulus Shi, Enhui Xu, Hui Yu, Ziqi Dynamical Systems Let $\mathbb{A}$ be an annulus in the plane $\mathbb R^2$ and $g:\mathbb{A}\rightarrow \mathbb{A}$ be a boundary components preserving homeomorphism which is distal and has no periodic points. In \cite{SXY}, the authors show that there is a continuous decomposition $\mathcal P$ of $\mathbb{A}$ into $g$-invariant circles such that all the restrictions of $g$ on them share a common irrational rotation number (also called the rotation number of $g$) and all these circles are linearly ordered by the inclusion relation on the sets of bounded components of their complements in $\mathbb R^2$. In this note, we show that if the decomposition $\mathcal P$ above has a continuous section, then $g$ can be linearized, that is it is topologically conjugate to a rigid rotation on $\mathbb{A}$. For every irrational number $α\in (0, 1)$, we show the existence of such a distal homeomorphism $g$ on $\mathbb{A}$ that it cannot be linearized and its rotation number is $α$. |
| title | Linearizations of periodic point free distal homeomorphisms on the annulus |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2411.18360 |