On \(g\)-measures in symbolic dynamics (Q980492)

In the theory of finite-valued stationary processes the class of Markov chains plays a fundamental role. In particular, the topological supports of finite \(n\)-step Markov chains form the class of shifts of finite type that are of first importance in symbolic dynamics. One of the natural generalizations of Markov chains is the class of finitary Markov pocesses described in [\textit{G.~Morvai} and \textit{B.~Weiss}, Probab. Theory Relat. Fields 132, No. 1, 1--12 (2005; Zbl 1061.62148)]. Much as a transition matrix is a basic notion in the study of Markov chains, for finitary Markov processes \((x_i)_{i\in\mathbb Z}\) one has the function that gives the conditional probability distribution of \(x_1\) given its past \((x_i)_{-\infty <i\leq 0}\). Functions that arise in this way belong to a class of functions that are studied in the paper under the name of residually locally constant \(g\)-functions. They can be extended by continuity to their maximal domain of definition. The study of their associated symbolic dynamics leads one to the so-called \(D\)-shifts [\textit{W.~Krieger}, On \(g\)-functions for subshifts. Dynamics and stochastics. Festschrift in honor of M. S. Keane. Selected papers based on the presentations at the conference `Dynamical systems, probability theory, and statistical mechanics', Eindhoven, The Netherlands, January 3--7, 2005, on the occasion of the 65th birthday of Mike S. Keane. Beachwood, OH: IMS, Institute of Mathematical Statistics. Institute of Mathematical Statistics. Lecture Notes - Monograph Series 48, 306--316 (2006; Zbl 1132.37009)]. In the paper the phenomena that can arise in residually locally constant and locally constant maximally defined \(g\)-functions on \(D\)-shifts, Markov shifts and synchronizing systems are studied with respect to future measures and \(g\)-measures.