Conductance and Noncommutative Dynamical Systems

Abstract

In this work, we introduce the notion of conductance in the context of Cuntz-Krieger C * -algebras. These algebras can be seen as a noncommutative version of topological Markov chains. Conductance is a useful notion in the theory of Markov chains to study the approach of a system to the equilibrium state. Our goal is twofold. On one hand, conductance can be used to measure the complexity of dynamical systems, complementing topological entropy. On the other hand, using C * -algebras, we can give a natural framework to analyze the path space of a finite graph associated to a Markov shift.