Lipschitz sub-actions for locally maximal hyperbolic sets of a $C^1$ flow
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866917704805384192 |
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| author | Su, Xifeng Thieullen, Philippe |
| author_facet | Su, Xifeng Thieullen, Philippe |
| contents | Livšic theorem for flows asserts that a Lipschitz observable that has zero mean average along every periodic orbit is necessarily a coboundary, that is the Lie derivative of a Lipschitz function smooth along the flow direction. The positive Livšic theorem bounds from below the observable by such a coboundary as soon as the mean average along every periodic orbit is non negative. Previous proofs give a Hölder coboundary. Assuming that the dynamics is given by a locally maximal hyperbolic flow, we show that the coboundary can be Lipschitz. We introduce a new tool: the Lax-Oleinik semigroup, inspired by Fathi's weak KAM theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_10135 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Lipschitz sub-actions for locally maximal hyperbolic sets of a $C^1$ flow Su, Xifeng Thieullen, Philippe Dynamical Systems Optimization and Control Livšic theorem for flows asserts that a Lipschitz observable that has zero mean average along every periodic orbit is necessarily a coboundary, that is the Lie derivative of a Lipschitz function smooth along the flow direction. The positive Livšic theorem bounds from below the observable by such a coboundary as soon as the mean average along every periodic orbit is non negative. Previous proofs give a Hölder coboundary. Assuming that the dynamics is given by a locally maximal hyperbolic flow, we show that the coboundary can be Lipschitz. We introduce a new tool: the Lax-Oleinik semigroup, inspired by Fathi's weak KAM theory. |
| title | Lipschitz sub-actions for locally maximal hyperbolic sets of a $C^1$ flow |
| topic | Dynamical Systems Optimization and Control |
| url | https://arxiv.org/abs/2205.10135 |