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Thermodynamic formalism for countable symbolic Markov chains. (English. Russian original) Zbl 0926.37009

Russ. Math. Surv. 53, No. 2, 245-344 (1998); translation from Usp. Mat. Nauk 53, No. 2, 3-106 (1998).
The thermodynamic formalism for symbolic Markov chains (also known as topological Markov chains) with finite alphabet has been well developed in the seventies. Since then many attempts have been made to extend the theory to countable Markov chains. The main problem here is non-compactness of the underlying spaces. Generally, countable Markov chains do not nearly have the same nice properties as finite ones. One has to impose rather tight restrictions on the transition matrix or potential functions to obtain reasonable results. Here most of the results are obtained for potential functions depending on \(n\) variables (i.e., constant on cylinders of length \(n\) with a fixed \(n\), the authors call them local functions). It is also standard that most of the interesting results are obtained for positively recurrent chains. Modulo these restrictions, the authors cover a wide range of properties.
The paper is partially a survey, but contains many new results as well. It discusses variational principle, Gibbs measures, dynamical zeta-function, and many other issues. The paper is well written and gives a good survey of the subject.

MSC:

37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
28D05 Measure-preserving transformations
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
37B10 Symbolic dynamics
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
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