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Spectral Triples on Thermodynamic Formalism and Dixmier Trace Representations of Gibbs: theory and examples

Leandro Cioletti, L. Y. Hataishi, Artur O. Lopes, M. Stadlbauer

Abstract

In this paper we study spectral triples and non-commutative expectations associated to expanding and weakly expanding maps. In order to do so, we generalize the Perron-Frobenius-Ruelle theorem and obtain a polynomial decay of the operator, which allows to prove differentiability of a dynamically defined $ζ$-function at its critical parameter. We then generalize Sharp's construction of spectral triples to this setting and provide criteria when the associated spectral metric is non-degenerate and when the non-commutative expectation of the spectral triple is colinear to the integration with respect to the associated equilibrium state from thermodynamic formalism. Due to our general setting, we are able to simultaneously analyse expanding maps on manifolds or connected fractals, subshifts of finite type as well as the Dyson model from statistical physics, which underlines the unifying character of noncommutative geometry. Furthermore, we derive an explicit representation of the $ζ$-function associated to a particular class of pathological continuous potentials, giving rise to examples where the representation as a non-commutative expectation via the associated zeta function holds, and others where it does not hold.

Spectral Triples on Thermodynamic Formalism and Dixmier Trace Representations of Gibbs: theory and examples

Abstract

In this paper we study spectral triples and non-commutative expectations associated to expanding and weakly expanding maps. In order to do so, we generalize the Perron-Frobenius-Ruelle theorem and obtain a polynomial decay of the operator, which allows to prove differentiability of a dynamically defined -function at its critical parameter. We then generalize Sharp's construction of spectral triples to this setting and provide criteria when the associated spectral metric is non-degenerate and when the non-commutative expectation of the spectral triple is colinear to the integration with respect to the associated equilibrium state from thermodynamic formalism. Due to our general setting, we are able to simultaneously analyse expanding maps on manifolds or connected fractals, subshifts of finite type as well as the Dyson model from statistical physics, which underlines the unifying character of noncommutative geometry. Furthermore, we derive an explicit representation of the -function associated to a particular class of pathological continuous potentials, giving rise to examples where the representation as a non-commutative expectation via the associated zeta function holds, and others where it does not hold.

Paper Structure

This paper contains 19 sections, 31 theorems, 217 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Asymptotics of Ruelle's operator
  3. Spectral Triples and their spectral metrics
  4. Spectral triples
  5. Spectral triples and their spectral metrics
  6. Subshifts of finite type
  7. A spectral triple associated to the Sierpiński gasket
  8. Dixmier Trace Representation of Gibbs Measures
  9. Proof of Theorem \ref{['theo:dixmier-for-Ruelle-expansive-and-expanding']}
  10. Examples and counterexamples
  11. The Dyson potential
  12. Potentials from Walters' family
  13. Dixmier Trace Representations for Equilibrium Measures of Topological Markov Shifts
  14. The Ruelle Operator applied to indicators of cylinders for a family of potentials
  15. Normalization of the Eigenvalue
  16. ...and 4 more sections

Key Result

Theorem 1.3

Assume that $T$ is topologically mixing, that $\mathcal{L}_a(\mathbf{1})=\mathbf{1}$ and that we are either in Case 1 or Case 2.

Figures (1)

  • Figure 1: An expanding map on the union of 4 Sierpiński gaskets

Theorems & Definitions (69)

  • Definition 1.1: Ruelle expansive
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1: Ruelle expansive
  • Example 2.2
  • Example 2.3
  • Proposition 2.4
  • Theorem 2.5
  • Remark 2.6
  • ...and 59 more