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

TL;DR

This paper develops a unified non-commutative geometric framework for analyzing spectral triples associated with expanding maps and thermodynamic formalism, including criteria for non-degeneracy and explicit zeta-function representations.

Contribution

It generalizes the Perron-Frobenius-Ruelle theorem, extends spectral triple constructions, and provides criteria for non-commutative expectations to align with equilibrium states across various dynamical systems.

Findings

Polynomial decay of operators enables differentiability of the zeta-function.

Criteria established for non-degenerate spectral metrics and colinearity of expectations.

Explicit examples show when non-commutative expectation representations hold or fail.

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 $\zeta$-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 $\zeta$-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.