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

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

Published 2018-04-04, updated 2020-07-17Version 2

In this paper we construct spectral triples $(A,H,D)$ on the symbolic space when the alphabet is finite. We describe some new results for the associated Dixmier trace representations for Gibbs probabilities (for potentials with less regularity than H\"older) and for a certain class of functions. The Dixmier trace representation can be expressed as the limit of a certain zeta function obtained from high order iterations of the Ruelle operator. Among other things we consider a class of examples where we can exhibit the explicit expression for the zeta function. We are also able to apply our reasoning for some parameters of the Dyson model (a potential on the symbolic space $\{-1,1\}^\mathbb{N}$) and for a certain class of observables. Nice results by R. Sharp, M.~Kesseb\"ohmer and T.~Samuel for Dixmier trace representations of Gibbs probabilities considered the case where the potential is of H\"older class. We also analyze a particular case of a pathological continuous potential where the Dixmier trace representation - via the associated zeta function - is not true.