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

In this paper I study properties of the generators $\triangle_\gamma$ of non-local Dirichlet forms $\mathcal{E}^\mu_\gamma$ 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 H\"older 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 $\triangle_\gamma$, 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 $\mathcal{E}^\mu_\gamma$-minimizing representative of any class $c\in H_{lc}(X_B)$.