|
Existence of Gibbs measures for countable Markov shifts
Author(s):
Omri
Sarig
Journal:
Proc. Amer. Math. Soc.
131
(2003),
1751-1758.
MSC (2000):
Primary 37A99, 37D35;
Secondary 37B10
Posted:
January 2, 2003
Retrieve article in:
PDF DVI PostScript
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:
-
- 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
Email:
sarig@maths.warwick.ac.uk
DOI:
10.1090/S0002-9939-03-06927-2
PII:
S 0002-9939(03)06927-2
Keywords:
Gibbs measures,
countable Markov shifts,
thermodynamic formalism
Received by editor(s):
October 5, 2001
Posted:
January 2, 2003
Additional Notes:
This work is part of a Tel-Aviv University dissertation.
Communicated by:
Michael Handel
Copyright of article:
Copyright
2003,
American Mathematical Society
|
Comments: webmaster@ams.org