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

Abstract In this paper I study properties of the generators $$\triangle _\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>▵</mml:mi> <mml:mi>γ</mml:mi> </mml:msub> </mml:math> of non-local Dirichlet forms $$\mathcal {E}^\mu _\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> <mml:mi>γ</mml:mi> <mml:mi>μ</mml:mi> </mml:msubsup> </mml:math> 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 $$\mu _\psi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>ψ</mml:mi> </mml:msub> </mml:math> associated to Hölder continuous potentials $$\psi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ψ</mml:mi> </mml:math> for one-sided shifts. I also define a cohomology $$H_{lc}(X_B)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mrow> <mml:mi>lc</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>B</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> for $$X_B$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>B</mml:mi> </mml:msub> </mml:math> which can be seen as dual to the homology of Bowen and Franks. Besides studying spectral properties of $$\triangle _\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>▵</mml:mi> <mml:mi>γ</mml:mi> </mml:msub> </mml:math> , I show that for $$\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> large enough (with sharp bounds depending on the diagram and the measure theoretic entropy $$h_{\mu _\psi }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>h</mml:mi> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>ψ</mml:mi> </mml:msub> </mml:msub> </mml:math> of $$\mu _\psi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>ψ</mml:mi> </mml:msub> </mml:math> ) there is a unique $$\mathcal {E}^\mu _\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> <mml:mi>γ</mml:mi> <mml:mi>μ</mml:mi> </mml:msubsup> </mml:math> -minimizing representative of any class $$c\in H_{lc}(X_B)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mrow> <mml:mi>lc</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>B</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> .