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

Abstract

In this paper we construct spectral triples (A, H, D) and describe the associated Dixmier trace representations of Gibbs measures of a large class of g-functions defined on symbolic space A N . We obtain results in both contexts of finite and infinite countable alphabets and provide examples with explicit computations. On finite alphabet setting one of the main contributions of this paper is to generalize this representations theorems to potentials in the Walters class, where in general the spectral gap property is absent. As an application we establish a noncommutative integral representation for the DLR-Gibbs measures of the Dyson model. Similar results are proved in the context of infinite countable alphabets, and Topological Markov Shifts, when the spectral gap property holds. In the last section, we make some remarks about this representation problem when A is an uncountable alphabet and discuss about its potential applications.