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Non-local Dirichlet forms, Gibbs measures, and a Hodge theorem for Cantor sets

Rodrigo Treviño

TL;DR

The paper develops a non-local Dirichlet-form framework on ultrametric Cantor sets arising from Bratteli diagrams, pairing these analytical objects with Gibbs measures from thermodynamic formalism. It introduces a locally constant cohomology dual to Bowen–Franks homology and proves a Cantor-set analogue of the Hodge theorem: for large γ every cohomology class has a unique harmonic representative in the Dirichlet-form domain. The work also delivers a detailed spectral analysis of the associated fractional Laplacians, including explicit eigenvalue spectra in several Bratteli-graph examples and sharp heat-kernel bounds, thereby connecting symbolic dynamics, non-local analysis, and topological invariants of Cantor sets. Overall, it provides a rigorous bridge between non-local Dirichlet forms on Bratteli-path spaces, Gibbs measures, and a Hodge-type decomposition on Cantor sets, with potential to identify diagrammatic invariants via spectral data.

Abstract

In this paper I study properties of the generators $\triangle_γ$ of non-local Dirichlet forms $\mathcal{E}^μ_γ$ 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 $μ_ψ$ associated to Hölder continuous potentials $ψ$ 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 $\triangle_γ$, I show that for $γ$ large enough (with sharp bounds depending on the diagram and the measure theoretic entropy $h_{μ_ψ}$ of $μ_ψ$) there is a unique harmonic representative of any class $c\in H_{lc}(X_B)$.

Non-local Dirichlet forms, Gibbs measures, and a Hodge theorem for Cantor sets

TL;DR

The paper develops a non-local Dirichlet-form framework on ultrametric Cantor sets arising from Bratteli diagrams, pairing these analytical objects with Gibbs measures from thermodynamic formalism. It introduces a locally constant cohomology dual to Bowen–Franks homology and proves a Cantor-set analogue of the Hodge theorem: for large γ every cohomology class has a unique harmonic representative in the Dirichlet-form domain. The work also delivers a detailed spectral analysis of the associated fractional Laplacians, including explicit eigenvalue spectra in several Bratteli-graph examples and sharp heat-kernel bounds, thereby connecting symbolic dynamics, non-local analysis, and topological invariants of Cantor sets. Overall, it provides a rigorous bridge between non-local Dirichlet forms on Bratteli-path spaces, Gibbs measures, and a Hodge-type decomposition on Cantor sets, with potential to identify diagrammatic invariants via spectral data.

Abstract

In this paper I study properties of the generators of non-local Dirichlet forms 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 associated to Hölder continuous potentials for one-sided shifts. I also define a cohomology for which can be seen as dual to the homology of Bowen and Franks. Besides studying spectral properties of , I show that for large enough (with sharp bounds depending on the diagram and the measure theoretic entropy of ) there is a unique harmonic representative of any class .
Paper Structure (20 sections, 22 theorems, 154 equations, 1 figure)

This paper contains 20 sections, 22 theorems, 154 equations, 1 figure.

Table of Contents

  1. Introduction and statement of results
  2. Statement of results
  3. Bratteli Diagrams and LF algebras
  4. Basic definitions
  5. Function spaces
  6. LF algebras and traces
  7. $LF$-algebras
  8. Traces
  9. Cohomology for $X_B$
  10. Gibbs measures
  11. Gibbs measures and cylinders
  12. Non-local Dirichlet forms
  13. Spectral properties of $\triangle_\gamma^\psi$
  14. Examples
  15. $A_1$
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $-\triangle_\gamma^\psi$ be the generator of $\mathcal{E}_\gamma^{\mu_\psi}$ on $W_{\psi,\gamma}^0$, where $\psi:X_B\rightarrow \mathbb{R}$ is a Hölder continuous function and $\mu_{\psi}$ is its unique Gibbs state. Then for $\gamma>d_\psi$:

Figures (1)

  • Figure 1: Three primitive simple diagrams which are related.

Theorems & Definitions (42)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2: Hodge theorem for Cantor sets
  • Lemma 2.1
  • proof
  • Corollary 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.1
  • proof
  • ...and 32 more