Tropical thermodynamic formalism
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908504845975552 |
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| author | Li, Zhiqiang Sun, Yiqing |
| author_facet | Li, Zhiqiang Sun, Yiqing |
| contents | We investigate the zero-temperature large deviation principle for equilibrium states in the context of distance-expanding maps. The logarithmic-type zero-temperature limit in the large deviation principle induces a tropical algebra structure, which motivates our study of the tropical adjoint Bousch operator $\mathcal{L}_A^{*}$ since the Bousch operator $\mathcal{L}_A$ is tropical linear and corresponds to the Ruelle operator $\mathcal{R}_A$.
We extend tropical functional analysis, define the adjoint operator $\mathcal{L}_A^{*}$ corresponding to $\mathcal{R}_A^{*}$, and establish the existence and generic uniqueness of tropical eigen-densities of $\mathcal{L}_A^{*}$. The Aubry set and the Mañé potential, both originating from weak KAM theory, serve as important tools in the representation of tropical eigen-densities.
We derive a sufficient condition for the large deviation principle which holds for a generic Hölder potential and establish a characterization theorem for the large deviation principle. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_10169 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tropical thermodynamic formalism Li, Zhiqiang Sun, Yiqing Dynamical Systems Primary: 37A99, Secondary: 15A80, 37C30, 37D35, 60F10 We investigate the zero-temperature large deviation principle for equilibrium states in the context of distance-expanding maps. The logarithmic-type zero-temperature limit in the large deviation principle induces a tropical algebra structure, which motivates our study of the tropical adjoint Bousch operator $\mathcal{L}_A^{*}$ since the Bousch operator $\mathcal{L}_A$ is tropical linear and corresponds to the Ruelle operator $\mathcal{R}_A$. We extend tropical functional analysis, define the adjoint operator $\mathcal{L}_A^{*}$ corresponding to $\mathcal{R}_A^{*}$, and establish the existence and generic uniqueness of tropical eigen-densities of $\mathcal{L}_A^{*}$. The Aubry set and the Mañé potential, both originating from weak KAM theory, serve as important tools in the representation of tropical eigen-densities. We derive a sufficient condition for the large deviation principle which holds for a generic Hölder potential and establish a characterization theorem for the large deviation principle. |
| title | Tropical thermodynamic formalism |
| topic | Dynamical Systems Primary: 37A99, Secondary: 15A80, 37C30, 37D35, 60F10 |
| url | https://arxiv.org/abs/2408.10169 |