The KMS Condition for the homoclinic equivalence relation and Gibbs probabilities

Abstract: D. Ruelle considered a general setting where he is able to characterize equilibrium states for H\"older potentials based on properties of conjugating homeomorphism in the so called Smale spaces. On this setting he also shows a relation of KMS states of $C^*$-algebras and equilibrium probabilities of Thermodynamic Formalism. A later paper by N. Haydn and D. Ruelle presents a shorter proof of this equivalence. Here we consider similar problems but now on the symbolic space $\Omega = {1,2,...,d}^{\mathbb{Z} - { 0 } }$ and the dynamics will be given by the shift $\tau$. In the case of potentials depending on a finite coordinates we will present a simplified proof of the equivalence mentioned above which is the main issue of the papers by D. Ruelle and N. Haydn. The class of conjugating homeomorphism is explicit and reduced to a minimal set of conditions. We also present with details (following D. Ruelle) the relation of these probabilities with the KMS dynamical $C^*$-state on the $C^*$-Algebra associated to the groupoid defined by the homoclinic equivalence relation. The topics presented here are not new but we believe the main ideas of the proof of the results by Ruelle and Haydn will be quite transparent in our exposition.