Equilibrium states for the classical Lorenz attractor and sectional-hyperbolic attractors in higher dimensions
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866910730021765120 |
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| author | Pacifico, Maria Jose Yang, Fan Yang, Jiagang |
| author_facet | Pacifico, Maria Jose Yang, Fan Yang, Jiagang |
| contents | It has long been conjectured that the classical Lorenz attractor supports a unique measure of maximal entropy. In this article, we give a positive answer to this conjecture and its higher-dimensional counterpart by considering the uniqueness of equilibrium states for Hölder continuous functions on a sectional-hyperbolic attractor $Λ$. We prove that in a $C^1$-open and dense family of vector fields (including the classical Lorenz attractor), if the point masses at singularities are not equilibrium states, then there exists a unique equilibrium state supported on $Λ$. In particular, there exists a unique measure of maximal entropy for the flow $X|_Λ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_10784 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Equilibrium states for the classical Lorenz attractor and sectional-hyperbolic attractors in higher dimensions Pacifico, Maria Jose Yang, Fan Yang, Jiagang Dynamical Systems It has long been conjectured that the classical Lorenz attractor supports a unique measure of maximal entropy. In this article, we give a positive answer to this conjecture and its higher-dimensional counterpart by considering the uniqueness of equilibrium states for Hölder continuous functions on a sectional-hyperbolic attractor $Λ$. We prove that in a $C^1$-open and dense family of vector fields (including the classical Lorenz attractor), if the point masses at singularities are not equilibrium states, then there exists a unique equilibrium state supported on $Λ$. In particular, there exists a unique measure of maximal entropy for the flow $X|_Λ$. |
| title | Equilibrium states for the classical Lorenz attractor and sectional-hyperbolic attractors in higher dimensions |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2209.10784 |