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Pressure and equilibrium states for countable state Markov shifts. (English) Zbl 1026.37020
The authors consider a directed graph \((V,E)\), where \(V\) is the set of vertices and \(E\) is the set of edges. Both sets \(V\) and \(E\) can be infinite but countable. The graph \(G\) is allowed to have parallel edges and it is irreducible. Then \(G\) defines the noncompact phase space \(X\) for a two-sided transitive Markov shift. In this setting, two definitions of topological pressure of continuous maps \(f:X\rightarrow \mathbb{R}\) are introduced, and it is proved that if \(f\) holds some distorsion properties, both definitions are equivalent. Additionally, an alternative equivalent definition of topological pressure for transitive Markov shifts whose set of vertices is finite is also given.
Later on, a variational principle for this notion is studied. Then, they introduce the notion of \(Z\)-recurrence of vertices of the graph, which is used to find the equilibrium states of the map \(f\).
A translation of these results for one-sided transitive Markov shifts is also studied, as well as distortion properties of continuous maps \(f:X\rightarrow \mathbb{R}\). The paper ends with an application of this theory to the Gauss map \(T:Y\rightarrow Y\), given by \(Tx=1/x-[1/x]\), where \(Y\) denotes the set of irrational numbers in \((0,1)\) and \([x]\) is the maximal integer less or equal than \(x\). The equilibrium state for the map \(f(x)=-\log |T'(x)|\) is characterized as a limiting measure obtained from measures supported on periodic points.

37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
37A05 Dynamical aspects of measure-preserving transformations
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
37B10 Symbolic dynamics
Full Text: DOI
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