Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Existence of Gibbs measures for countable Markov shifts

Author: Omri Sarig
Journal: Proc. Amer. Math. Soc. 131 (2003), 1751-1758
MSC (2000): Primary 37A99, 37D35; Secondary 37B10
Published electronically: January 2, 2003
MathSciNet review: 1955261
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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. Urbanski (2001) who showed that this condition is sufficient.

References [Enhancements On Off] (What's this?)

  • 1. Aaronson, J.: An introduction to infinite Ergodic Theory, Math. Surv. and Mono. 50 (1997), AMS. MR 99d:28025
  • 2. Aaronson, J., Denker, M., Urbanski: Ergodic theory of Markov fibered systems and parabolic rational maps. Trans. Am. Math. Soc. 337 (1993), 495-548. MR 94g:58116
  • 3. Aaronson, J., Denker, M.: Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps. Stoch. Dyn. 1 (2001), no. 2, 193-237. MR 2002h:37014
  • 4. Bowen, R.: Equilibrium states and the theory of Anosov diffeomorphisms. Lect. Notes in Math. 470, Springer Verlag (1975). MR 56:1364
  • 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.
  • 6. Gurevic, B.M.: Topological entropy for denumerable Markov chains. Dokl. Akad. Nauk. SSSR 187 (1969); English Transl. in Soviet Math. Dokl. 10 (1969), 911-915. MR 41:7767
  • 7. Mauldin, R.D., Urbanski, M.: Gibbs states on the symbolic space over an infinite alphabet. Israel J. Maths. 125 (2001), 93-130. MR 2002k:37048
  • 8. Ruelle, D.: Thermodynamic formalism. Encyclopedia of Mathematics and its Applications 5, Addison-Wesley (1978). MR 80g:82017
  • 9. Salama, I.: On the recurrence of countable topological Markov chains. Pacific J. Math. 134 (1988), 325-341. Errata Pac. J. Math. 140 (1989), 397. MR 90k:54055
  • 10. Sarig, O.: Thermodynamic Formalism for Countable Markov Shifts. Ergod. Th. Dynam. Sys. 19 (1999), 1565-1593. MR 2000m:37009
  • 11. Sarig, O.: Thermodynamic formalism for null recurrent potentials. Israel J. Math. 121 (2001), 285-311. MR 2001m:37059
  • 12. Sarig, O.: Phase transitions for countable Markov shifts. Commun. Math. Phys. 217 (2001), 555-577. MR 2002b:37040
  • 13. Sarig, O.: On an example with topological pressure which is not analytic. C.R. Acad. Sci. Serie I: Math. 330 (2000), 311-315. MR 2000m:37020

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 37A99, 37D35, 37B10

Retrieve articles in all journals with MSC (2000): 37A99, 37D35, 37B10

Additional Information

Omri Sarig
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, England

Keywords: Gibbs measures, countable Markov shifts, thermodynamic formalism
Received by editor(s): October 5, 2001
Published electronically: January 2, 2003
Additional Notes: This work is part of a Tel-Aviv University dissertation.
Communicated by: Michael Handel
Article copyright: © Copyright 2003 American Mathematical Society