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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.22742 |
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| _version_ | 1866910217830137856 |
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| author | Treviño, Rodrigo |
| author_facet | Treviño, Rodrigo |
| contents | In this paper I study properties of the generators $\triangle_γ$ of non-local Dirichlet forms $\mathcal{E}^μ_γ$ on ultrametric spaces which are the path space of simple stationary Bratteli diagrams. The measures used to define the Dirichlet forms are taken to be the Gibbs measures $μ_ψ$ associated to Hölder continuous potentials $ψ$ for one-sided shifts. I also define a cohomology $H_{lc}(X_B)$ for $X_B$ which can be seen as dual to the homology of Bowen and Franks. Besides studying spectral properties of $\triangle_γ$, I show that for $γ$ large enough (with sharp bounds depending on the diagram and the measure theoretic entropy $h_{μ_ψ}$ of $μ_ψ$) there is a unique $\mathcal{E}^μ_γ$-minimizing representative of any class $c\in H_{lc}(X_B)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_22742 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-local Dirichlet forms, Gibbs measures, and a cohomological Dirichlet principle for Cantor sets Treviño, Rodrigo Dynamical Systems Mathematical Physics Analysis of PDEs Operator Algebras In this paper I study properties of the generators $\triangle_γ$ of non-local Dirichlet forms $\mathcal{E}^μ_γ$ on ultrametric spaces which are the path space of simple stationary Bratteli diagrams. The measures used to define the Dirichlet forms are taken to be the Gibbs measures $μ_ψ$ associated to Hölder continuous potentials $ψ$ for one-sided shifts. I also define a cohomology $H_{lc}(X_B)$ for $X_B$ which can be seen as dual to the homology of Bowen and Franks. Besides studying spectral properties of $\triangle_γ$, I show that for $γ$ large enough (with sharp bounds depending on the diagram and the measure theoretic entropy $h_{μ_ψ}$ of $μ_ψ$) there is a unique $\mathcal{E}^μ_γ$-minimizing representative of any class $c\in H_{lc}(X_B)$. |
| title | Non-local Dirichlet forms, Gibbs measures, and a cohomological Dirichlet principle for Cantor sets |
| topic | Dynamical Systems Mathematical Physics Analysis of PDEs Operator Algebras |
| url | https://arxiv.org/abs/2510.22742 |