Non-local Dirichlet forms, Gibbs measures, and a cohomological Dirichlet principle for Cantor sets

Abstract

In this paper I study properties of the generators ${△}_{\gamma }$ of non-local Dirichlet forms ${\mathcal{E}}_{\gamma }^{\mu }$ 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 ${\mu }_{\psi }$ associated to Hölder continuous potentials $\psi$ 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 ${△}_{\gamma }$, I show that for $\gamma$ large enough (with sharp bounds depending on the diagram and the measure theoretic entropy $h_{{\mu }_{\psi }}$ of ${\mu }_{\psi }$) there is a unique ${\mathcal{E}}_{\gamma }^{\mu }$-minimizing representative of any class $c\in H_{lc}(X_B)$.

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