Multidimensional non-uniform hyperbolicity, robust exponential mixing and the basin problem
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916688636674048 |
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| author | Araujo, Vitor Pinheiro, Vilton |
| author_facet | Araujo, Vitor Pinheiro, Vilton |
| contents | We show that the ergodic, topological and geometric basins coincide for hyperbolic dominated ergodic $cu$-Gibbs states, solving the ``basin problem'' for a wide class of non-uniformly hyperbolic systems.
We obtain robust examples of exponential mixing physical measures for systems with multidimensional nonuniform hyperbolic dominated splitting, without uniformly expanding or contracting subbundles.
Both results are a consequence of extending the construction of Gibbs-Markov-Young structures from partial hyperbolic systems to systems with only a dominated splitting, using the existence of an ``improved hyperbolic block'', with respect to Pesin's Nonuniform Hyperbolic Theory, for hyperbolic dominated measures of smooth maps, obtained through hyperbolic times and associated ``coherent schedules'' introduced by one of the coauthors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_10264 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multidimensional non-uniform hyperbolicity, robust exponential mixing and the basin problem Araujo, Vitor Pinheiro, Vilton Dynamical Systems Primary: 37A25, 37C70. Secondary: 37C40 We show that the ergodic, topological and geometric basins coincide for hyperbolic dominated ergodic $cu$-Gibbs states, solving the ``basin problem'' for a wide class of non-uniformly hyperbolic systems. We obtain robust examples of exponential mixing physical measures for systems with multidimensional nonuniform hyperbolic dominated splitting, without uniformly expanding or contracting subbundles. Both results are a consequence of extending the construction of Gibbs-Markov-Young structures from partial hyperbolic systems to systems with only a dominated splitting, using the existence of an ``improved hyperbolic block'', with respect to Pesin's Nonuniform Hyperbolic Theory, for hyperbolic dominated measures of smooth maps, obtained through hyperbolic times and associated ``coherent schedules'' introduced by one of the coauthors. |
| title | Multidimensional non-uniform hyperbolicity, robust exponential mixing and the basin problem |
| topic | Dynamical Systems Primary: 37A25, 37C70. Secondary: 37C40 |
| url | https://arxiv.org/abs/2504.10264 |