Katok's entropy conjecture near real and complex hyperbolic metrics
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909854794252288 |
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| author | Humbert, Tristan |
| author_facet | Humbert, Tristan |
| contents | We show that, given a real or complex hyperbolic metric $g_0$ on a closed manifold $M$ of dimension $n\geq 3$, there exists a neighborhood $\mathcal U$ of $g_0$ in the space of negatively curved metrics such that for any $g\in \mathcal U$, the topological entropy and Liouville entropy of $g$ coincide if and only if $g$ and $g_0$ are homothetic. This provides a partial answer to Katok's entropy rigidity conjecture. As a direct consequence of our theorem, we obtain a local rigidity result for the hyperbolic rank and for metrics with $C^2$ Anosov foliations near complex hyperbolic metrics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_11197 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Katok's entropy conjecture near real and complex hyperbolic metrics Humbert, Tristan Dynamical Systems We show that, given a real or complex hyperbolic metric $g_0$ on a closed manifold $M$ of dimension $n\geq 3$, there exists a neighborhood $\mathcal U$ of $g_0$ in the space of negatively curved metrics such that for any $g\in \mathcal U$, the topological entropy and Liouville entropy of $g$ coincide if and only if $g$ and $g_0$ are homothetic. This provides a partial answer to Katok's entropy rigidity conjecture. As a direct consequence of our theorem, we obtain a local rigidity result for the hyperbolic rank and for metrics with $C^2$ Anosov foliations near complex hyperbolic metrics. |
| title | Katok's entropy conjecture near real and complex hyperbolic metrics |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2409.11197 |