On the relation between Gibbs and -measures

Abstract

Thermodynamic formalism, the theory of equilibrium states, is studied both in dynamical systems and probability theory. Various closely related notions have been developed: e.g. Dobrushin–Lanford–Ruelle Gibbs, Bowen–Gibbs and $g$-measures. We discuss the relation between Gibbs and $g$-measures in a one-dimensional context. Often $g$-measures are also Gibbs, but recently an example to the contrary has been presented. In this paper we discuss exactly when a $g$-measure is Gibbs and how this relates to notions such as uniqueness and reversibility of $g$-measures.