Upper semi-continuity of metric entropy for diffeomorphisms with dominated splitting
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866910761042837504 |
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| author | Luo, Chiyi Ma, Wenhui Zhao, Yun |
| author_facet | Luo, Chiyi Ma, Wenhui Zhao, Yun |
| contents | For a $C^{r}$ $(r>1)$ diffeomorphism on a compact manifold that admits a dominated splitting, this paper establishes the upper semi-continuity of the entropy map. More precisely, this paper establishes the upper semi-continuity of the entropy map in the following two cases:
(1) if a sequence of invariant measures has only positive Lyapunov exponents along a sub-bundle and non-positive Lyapunov exponents along another sub-bundle, then the upper limit of their metric entropies is less than or equal to the entropy of the limiting measure; (2) if an invariant measure has positive Lyapunov exponents along a sub-bundle and non-positive Lyapunov exponents along another sub-bundle, then the entropy map is upper semi-continuous at this measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_04953 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Upper semi-continuity of metric entropy for diffeomorphisms with dominated splitting Luo, Chiyi Ma, Wenhui Zhao, Yun Dynamical Systems For a $C^{r}$ $(r>1)$ diffeomorphism on a compact manifold that admits a dominated splitting, this paper establishes the upper semi-continuity of the entropy map. More precisely, this paper establishes the upper semi-continuity of the entropy map in the following two cases: (1) if a sequence of invariant measures has only positive Lyapunov exponents along a sub-bundle and non-positive Lyapunov exponents along another sub-bundle, then the upper limit of their metric entropies is less than or equal to the entropy of the limiting measure; (2) if an invariant measure has positive Lyapunov exponents along a sub-bundle and non-positive Lyapunov exponents along another sub-bundle, then the entropy map is upper semi-continuous at this measure. |
| title | Upper semi-continuity of metric entropy for diffeomorphisms with dominated splitting |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2412.04953 |