Existence of Gibbs measures for countable Markov shifts

Sarig, Omri

doi:10.1090/S0002-9939-03-06927-2

Existence of Gibbs measures for countable Markov shifts
HTML articles powered by AMS MathViewer

by Omri Sarig

Proc. Amer. Math. Soc. 131 (2003), 1751-1758

DOI: https://doi.org/10.1090/S0002-9939-03-06927-2

Published electronically: January 2, 2003

PDF | Request permission

Abstract:

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 D. Mauldin and M. Urbański (2001) who showed that this condition is sufficient.

References

Jon Aaronson, An introduction to infinite ergodic theory, Mathematical Surveys and Monographs, vol. 50, American Mathematical Society, Providence, RI, 1997. MR 1450400, DOI 10.1090/surv/050
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. MR 1107025, DOI 10.1090/S0002-9947-1993-1107025-2
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. MR 1840194, DOI 10.1142/S0219493701000114
Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR 0442989, DOI 10.1007/BFb0081279
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.
B. M. Gurevič, Topological entropy of a countable Markov chain, Dokl. Akad. Nauk SSSR 187 (1969), 715–718 (Russian). MR 0263162
R. Daniel Mauldin and Mariusz Urbański, Gibbs states on the symbolic space over an infinite alphabet, Israel J. Math. 125 (2001), 93–130. MR 1853808, DOI 10.1007/BF02773377
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. MR 511655
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. MR 1023792, DOI 10.2140/pjm.1989.140.397
Omri M. Sarig, Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems 19 (1999), no. 6, 1565–1593. MR 1738951, DOI 10.1017/S0143385799146820
Omri M. Sarig, Thermodynamic formalism for null recurrent potentials, Israel J. Math. 121 (2001), 285–311. MR 1818392, DOI 10.1007/BF02802508
Omri M. Sarig, Phase transitions for countable Markov shifts, Comm. Math. Phys. 217 (2001), no. 3, 555–577. MR 1822107, DOI 10.1007/s002200100367
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). MR 1753300, DOI 10.1016/S0764-4442(00)00189-0