Entropy formula of folding type for $C^{1+α}$ maps
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866908633547145216 |
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| author | Liao, Gang Wang, Shirou |
| author_facet | Liao, Gang Wang, Shirou |
| contents | In the study of non-equilibrium statistical mechanics, Ruelle derived explicit formulae for entropy production of smooth dynamical systems. The vanishing or strict positivity of entropy production is determined by the {\it entropy formula of folding type} \[h_μ(f)= F_μ(f)-\displaystyle\int\sum\nolimits_{λ_i(x)<0} λ_i(x)dμ(x), \] which relates the metric entropy, folding entropy and negative Lyapunov exponents. This paper establishes the formula for all inverse SRB measures of $C^{1+α}$ maps, including those with degeneracy (i.e., zero Jacobian). More specifically, we establish the equivalence that $μ$ is an inverse SRB measure if and only if the folding-type entropy formula holds and the Jacobian series is integrable. To overcome the degeneracy, we develop Pesin theory for general $C^{1+α}$ maps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_18344 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Entropy formula of folding type for $C^{1+α}$ maps Liao, Gang Wang, Shirou Dynamical Systems Mathematical Physics 37C40 (Primary), 37A60, 37D25 (Secondary) In the study of non-equilibrium statistical mechanics, Ruelle derived explicit formulae for entropy production of smooth dynamical systems. The vanishing or strict positivity of entropy production is determined by the {\it entropy formula of folding type} \[h_μ(f)= F_μ(f)-\displaystyle\int\sum\nolimits_{λ_i(x)<0} λ_i(x)dμ(x), \] which relates the metric entropy, folding entropy and negative Lyapunov exponents. This paper establishes the formula for all inverse SRB measures of $C^{1+α}$ maps, including those with degeneracy (i.e., zero Jacobian). More specifically, we establish the equivalence that $μ$ is an inverse SRB measure if and only if the folding-type entropy formula holds and the Jacobian series is integrable. To overcome the degeneracy, we develop Pesin theory for general $C^{1+α}$ maps. |
| title | Entropy formula of folding type for $C^{1+α}$ maps |
| topic | Dynamical Systems Mathematical Physics 37C40 (Primary), 37A60, 37D25 (Secondary) |
| url | https://arxiv.org/abs/2411.18344 |