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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.

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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