The generalized Markov measure as an equilibrium state

Abstract

In this paper, we continue development of the theory of contractive Markov systems (CMS) initiated in \cite{Wer1}. Also, this work can be seen as a small contribution to the theory of equilibrium states. We construct an energy function on the code space, using the coding map from \cite{Wer3}, and show that the generalized Markov measure associated with an irreducible CMS is a unique equilibrium state for this energy function if the vertex sets form an open partition of the state space of the CMS and the restrictions of the probability functions on their vertex sets are Dini-continuous and bounded away from zero.