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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2409.05857 |
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| _version_ | 1866929492693352448 |
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| author | O'Hare, Thomas Aloysius |
| author_facet | O'Hare, Thomas Aloysius |
| contents | Let $f,g$ be $C^2$ area-preserving Anosov diffeomorphisms on $\mathbb{T}^2$ which are topologically conjugate by a homeomorphism $h$ ($hf=gh$). We assume that the Jacobian periodic data of $f$ and $g$ are matched by $h$ for all points of some large period $N\in\mathbb{N}$. We show that $f$ and $g$ are ``approximately smoothly conjugate." That is, there exists a $C^{1+α}$ diffeomorphism $\overline{h}_N$ such that $h$ and $\overline{h}_N$ are $C^0$ exponentially close in $N$, and $f$ and $f_N:=\overline{h}_N^{-1}g\overline{h}_N$ are $C^1$ exponentially close in $N$. Moreover, the rates of convergence are uniform among different $f,g$ in a $C^2$ bounded set of Anosov diffeomorphisms. The main idea in constructing $\overline{h}_N$ is to do a ``weighted holonomy" construction, and the main technical tool in obtaining our estimates is a uniform effective version of Bowen's equidistribution theorem of weighted discrete orbits to the SRB measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_05857 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finite Periodic Data Rigidity For Two-Dimensional Area-Preserving Anosov Diffeomorphisms O'Hare, Thomas Aloysius Dynamical Systems Let $f,g$ be $C^2$ area-preserving Anosov diffeomorphisms on $\mathbb{T}^2$ which are topologically conjugate by a homeomorphism $h$ ($hf=gh$). We assume that the Jacobian periodic data of $f$ and $g$ are matched by $h$ for all points of some large period $N\in\mathbb{N}$. We show that $f$ and $g$ are ``approximately smoothly conjugate." That is, there exists a $C^{1+α}$ diffeomorphism $\overline{h}_N$ such that $h$ and $\overline{h}_N$ are $C^0$ exponentially close in $N$, and $f$ and $f_N:=\overline{h}_N^{-1}g\overline{h}_N$ are $C^1$ exponentially close in $N$. Moreover, the rates of convergence are uniform among different $f,g$ in a $C^2$ bounded set of Anosov diffeomorphisms. The main idea in constructing $\overline{h}_N$ is to do a ``weighted holonomy" construction, and the main technical tool in obtaining our estimates is a uniform effective version of Bowen's equidistribution theorem of weighted discrete orbits to the SRB measure. |
| title | Finite Periodic Data Rigidity For Two-Dimensional Area-Preserving Anosov Diffeomorphisms |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2409.05857 |