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

In this paper I study properties of the generators \(\triangle _\gamma \) ${▵}_{\gamma }$ of non-local Dirichlet forms \(\mathcal {E}^\mu _\gamma \) ${\mathcal{E}}_{\gamma }^{\mu }$ 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 \(\mu _\psi \) ${\mu }_{\psi }$ associated to Hölder continuous potentials \(\psi \) $\psi$ for one-sided shifts. I also define a cohomology \(H_{lc}(X_B)\) $H_{\mathit{lc}}(X_B)$ for \(X_B\) $X_B$ which can be seen as dual to the homology of Bowen and Franks. Besides studying spectral properties of \(\triangle _\gamma \) ${▵}_{\gamma }$ , I show that for \(\gamma \) $\gamma$ large enough (with sharp bounds depending on the diagram and the measure theoretic entropy \(h_{\mu _\psi }\) $h_{{\mu }_{\psi }}$ of \(\mu _\psi \) ${\mu }_{\psi }$ ) there is a unique \(\mathcal {E}^\mu _\gamma \) ${\mathcal{E}}_{\gamma }^{\mu }$ -minimizing representative of any class \(c\in H_{lc}(X_B)\) $c\in H_{\mathit{lc}}(X_B)$ .