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Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps. (English) Zbl 1037.37005
Let $$(\Sigma ,\sigma )$$ be a (one-sided) topological Markov shift with countable set of states $$S$$ with transition matrix $$A=(t_{ij})_{S\times S}$$ ($$t_{ij}$$ being zero or one), i.e., $\Sigma =\{ (x_0,x_1,\dots )\in S^{\mathbb N\cup \{0\}}:t_{x_ix_{i+1}}=1 \text{ for all i}\},$ and $$\sigma :\Sigma\to\Sigma$$ is the left shift. From results of W. Parry, D. Ruelle, and P. Walters it follows that, if $$S$$ is finite and $$\sigma$$ is topologically transitive then there exists exactly one maximal measure and, for $$\phi :\Sigma\to\mathbb R$$ with summable variations $$\sum\text{ var}_n(\phi )<\infty$$, the potential $$\phi$$ has exactly one equilibrium measure. As B. M. Gurevich and S. V. Savchenko [Russ. Math. Surv. 53, 245–344 (1998; Zbl 0926.37009)] showed, in the case when $$S$$ is infinite, there exists a maximal measure if and only if $$A$$ is $$R$$-recurrent and $$R$$-positive and they proved the uniqueness of equilibrium measures of potentials of the form $$\phi (x)=\phi (x_0,\dots ,x_N)$$.
The paper gives a Ruelle-type generalization of the last result for $$\phi$$ depending on an infinite number of coordinates such that $$\sum_{n\geq 2}\text{ var}_n\phi <\infty$$. The generalization obtained is applied to certain (non-Markovian) multidimensional piecewise expanding maps by using their Markov diagrams.

##### MSC:
 37A30 Ergodic theorems, spectral theory, Markov operators 37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37A05 Dynamical aspects of measure-preserving transformations 37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems 37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
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