Pressure and equilibrium states for countable state Markov shifts
type: Item
instance of: scholarly article
title: Pressure and equilibrium states for countable state Markov shifts
zbMATH Open document ID: 1026.37020
DOI: 10.1007/BF02785859
author: Doris Fiebig
author: Ulf-Rainer Fiebig
author: Michiko Yuri
published in: Israel Journal of Mathematics
publication date: 2003-11-30T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 37D35
Mathematics Subject Classification ID: 37A05
Mathematics Subject Classification ID: 37A35
Mathematics Subject Classification ID: 37B10
zbMATH DE Number: 1850611
zbMATH Keywords: countable Markov shifts
zbMATH Keywords: topological pressure
zbMATH Keywords: equilibrium states
zbMATH Keywords: coding theory
reviewed by: Jose S. Canovas
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1850611