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

TL;DR

This paper explores non-local Dirichlet forms on ultrametric spaces derived from Bratteli diagrams, analyzing spectral properties and cohomological aspects related to Gibbs measures and potential functions.

Contribution

It introduces a cohomology theory for ultrametric spaces dual to Bowen and Franks homology, and establishes bounds for unique minimizers in the associated Dirichlet forms.

Findings

Spectral properties of the generators are characterized.

Unique minimizers exist for large enough gamma, with bounds depending on entropy.

Cohomology H_{lc}(X_B) is shown to be dual to Bowen and Franks homology.

Abstract

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 $c\in H_{lc}(X_B)$.