KMS States, Entropy, and a Variational Principle for Pressure

Abstract

We relate the concepts of entropy and pressure to that of KMS states for $C^*$-algebras. Several different definitions of entropy are known in our days: the one we present here is quite natural, extending the usual one for Dynamical Systems in Thermodynamic Formalism Theory, being basically obtained from transfer operators (also called Ruelle operators) and having the advantage of being very easily introduced. We also present a concept of pressure as a min-max principle. Later on, we consider the concept of a KMS state as an equilibrium state for a potential, in the context of $C^*$-algebras, and we show that there is a relation between equilibrium measures and KMS states for certain algebras arising from a continuous transformation.