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

In this paper I study properties of the generators _γ ▵ γ of non-local Dirichlet forms E^μ _γ E γ μ 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 μ ψ associated to Hölder continuous potentials ψ ψ for one-sided shifts. I also define a cohomology Hlc(XB) H lc ( X B ) for XB X B which can be seen as dual to the homology of Bowen and Franks. Besides studying spectral properties of _γ ▵ γ , I show that for γ γ large enough (with sharp bounds depending on the diagram and the measure theoretic entropy hμ _ψ h μ ψ of μ _ψ μ ψ ) there is a unique E^μ _γ E γ μ -minimizing representative of any class c∈ Hlc(XB) c ∈ H lc ( X B ) .