Sarig, Omri Existence of Gibbs measures for countable Markov shifts. (English) Zbl 1009.37003 Proc. Am. Math. Soc. 131, No. 6, 1751-1758 (2003). Summary: We prove that a potential with summable variations and finite pressure on a topologically mixing countable Markov shift has a Gibbs measure iff the transition matrix satisfies the big images and preimages property. This strengthens a result of R. D. Mauldin and M. Urbánski [Isr. J. Math. 125, 93-130 (2001; Zbl 1016.37005)] who showed that this condition is sufficient. Cited in 2 ReviewsCited in 95 Documents MSC: 37A30 Ergodic theorems, spectral theory, Markov operators 37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems 37B10 Symbolic dynamics Keywords:Gibbs measures; countable Markov shifts; thermodynamic formalism Citations:Zbl 1016.37005 PDFBibTeX XMLCite \textit{O. Sarig}, Proc. Am. Math. Soc. 131, No. 6, 1751--1758 (2003; Zbl 1009.37003) Full Text: DOI References: [1] Jon Aaronson, An introduction to infinite ergodic theory, Mathematical Surveys and Monographs, vol. 50, American Mathematical Society, Providence, RI, 1997. · Zbl 0882.28013 [2] Jon Aaronson, Manfred Denker, and Mariusz Urbański, Ergodic theory for Markov fibred systems and parabolic rational maps, Trans. Amer. Math. Soc. 337 (1993), no. 2, 495 – 548. · Zbl 0789.28010 [3] Jon Aaronson and Manfred Denker, Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps, Stoch. Dyn. 1 (2001), no. 2, 193 – 237. · Zbl 1039.37002 · doi:10.1142/S0219493701000114 [4] Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. · Zbl 0308.28010 [5] Buzzi, J., Sarig, O: Uniqueness of equilibrium measures for countable Markov shifts and multi-dimensional piecewise expanding maps. To appear in Erg. Thy. Dynam. Syst. · Zbl 1037.37005 [6] B. M. Gurevič, Topological entropy of a countable Markov chain, Dokl. Akad. Nauk SSSR 187 (1969), 715 – 718 (Russian). [7] R. Daniel Mauldin and Mariusz Urbański, Gibbs states on the symbolic space over an infinite alphabet, Israel J. Math. 125 (2001), 93 – 130. · Zbl 1016.37005 · doi:10.1007/BF02773377 [8] David Ruelle, Thermodynamic formalism, Encyclopedia of Mathematics and its Applications, vol. 5, Addison-Wesley Publishing Co., Reading, Mass., 1978. The mathematical structures of classical equilibrium statistical mechanics; With a foreword by Giovanni Gallavotti and Gian-Carlo Rota. · Zbl 0401.28016 [9] Ibrahim A. Salama, Corrections to: ”Topological entropy and recurrence of countable chains” [Pacific J. Math. 134 (1988), no. 2, 325 – 341; MR0961239 (90d:54076)], Pacific J. Math. 140 (1989), no. 2, 397. · Zbl 0619.54031 [10] Omri M. Sarig, Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems 19 (1999), no. 6, 1565 – 1593. · Zbl 0994.37005 · doi:10.1017/S0143385799146820 [11] Omri M. Sarig, Thermodynamic formalism for null recurrent potentials, Israel J. Math. 121 (2001), 285 – 311. · Zbl 0992.37025 · doi:10.1007/BF02802508 [12] Omri M. Sarig, Phase transitions for countable Markov shifts, Comm. Math. Phys. 217 (2001), no. 3, 555 – 577. · Zbl 1007.37018 · doi:10.1007/s002200100367 [13] Omri M. Sarig, On an example with a non-analytic topological pressure, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 4, 311 – 315 (English, with English and French summaries). · Zbl 0992.37002 · doi:10.1016/S0764-4442(00)00189-0 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.