Continuity properties of ergodic measures of maximal entropy for $C^r$ surface diffeomorphisms
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912733171023872 |
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| author | Buzzi, Jérôme Luo, Chiyi Yang, Dawei |
| author_facet | Buzzi, Jérôme Luo, Chiyi Yang, Dawei |
| contents | Let $f$ be a $C^r$ surface diffeomorphism with large entropy (more precisely, $h_{\rm top}(f)>λ_{\min}(f)/{r}$). Then the number of ergodic measures of maximal entropy is upper semicontinuous at $f$. This generalizes the $C^\infty$ case studied in \cite{BCS22}, answering Question 1.9 there.
Moreover, the number of such measures is locally constant if and only if every ergodic measure of maximal entropy of $f$ admits an ergodic continuation under small perturbations. In this case, the accumulation points of ergodic measures of maximal entropy are themselves ergodic. These facts are new, even in the $C^\infty$ case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_19658 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Continuity properties of ergodic measures of maximal entropy for $C^r$ surface diffeomorphisms Buzzi, Jérôme Luo, Chiyi Yang, Dawei Dynamical Systems Let $f$ be a $C^r$ surface diffeomorphism with large entropy (more precisely, $h_{\rm top}(f)>λ_{\min}(f)/{r}$). Then the number of ergodic measures of maximal entropy is upper semicontinuous at $f$. This generalizes the $C^\infty$ case studied in \cite{BCS22}, answering Question 1.9 there. Moreover, the number of such measures is locally constant if and only if every ergodic measure of maximal entropy of $f$ admits an ergodic continuation under small perturbations. In this case, the accumulation points of ergodic measures of maximal entropy are themselves ergodic. These facts are new, even in the $C^\infty$ case. |
| title | Continuity properties of ergodic measures of maximal entropy for $C^r$ surface diffeomorphisms |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2412.19658 |