# American Institute of Mathematical Sciences

January  2014, 8(1): 1-14. doi: 10.3934/jmd.2014.8.1

## On the work of Sarig on countable Markov chains and thermodynamic formalism

 1 Department of Mathematics, McAllister Building, Pennsylvania State University, University Park, PA 16802

Published  July 2014

The paper is a nontechnical survey and is aimed to illustrate Sarig's profound contributions to statistical physics and in particular, thermodynamic formalism for countable Markov shifts. I will discuss some of Sarig's work on characterization of existence of Gibbs measures, existence and uniqueness of equilibrium states as well as phase transitions for Markov shifts on a countable set of states.
Citation: Yakov Pesin. On the work of Sarig on countable Markov chains and thermodynamic formalism. Journal of Modern Dynamics, 2014, 8 (1) : 1-14. doi: 10.3934/jmd.2014.8.1
##### References:
 [1] J. Aaronson and M. Denker, Local limit theorems for Gibbs-Markov maps,, Stochastics Dyn., 1 (2001), 193. doi: 10.1142/S0219493701000114. Google Scholar [2] J. Aaronson, M. Denker and M. Urbański, Ergodic theory for Markov fibered systems and parabolic rational maps,, Trans. AMS, 337 (1993), 495. doi: 10.1090/S0002-9947-1993-1107025-2. Google Scholar [3] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,, Second revised edition, (2008). Google Scholar [4] J. Buzzi and O. Sarig, Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps,, Ergod. Th. and Dyn. Syst., 23 (2003), 1383. doi: 10.1017/S0143385703000087. Google Scholar [5] V. Cyr and O. Sarig, Spectral gap and transience for Ruelle operators on countable Markov shifts,, Comm. Math. Phys., 292 (2009), 637. doi: 10.1007/s00220-009-0891-4. Google Scholar [6] R. Dobrušin, Description of a random field by means of conditional probabilities and conditions for its regularity,, Teor. Veroyatnoistei i Primenenia, 13 (1968), 201. Google Scholar [7] R. Dobrušin, The problem of uniqueness of a Gibbsian random field and the problem of phase transitions,, Functional Anal. Appl., 2 (1968), 302. Google Scholar [8] M. Gordin, On the Central Limit Theorem for stationary processes,, (Russian) Doklady Akademii Nauk SSSR, 188 (1969), 739. Google Scholar [9] B. M. Gurevič, Topological entropy for denumerable Markov chains,, Dokl. Acad. Nauk SSSR, 187 (1969), 715. Google Scholar [10] B. M. Gurevič, Shift entropy and Markov measures in the space of paths of a countable graph,, Dokl. Acad. Nauk SSSR, 192 (1970), 963. Google Scholar [11] B. M. Gurevič, A variational characterization of one-dimensional countable state Gibbs random fields,, Z. Wahrsch. Verw. Gebiete, 68 (1984), 205. doi: 10.1007/BF00531778. Google Scholar [12] B. M. Gurevič and S. V. Savchenko, Thermodynamic formalism for symbolic Markov chainswith a countable number of states,, Uspehi. Mat. Nauk, 53 (1998), 3. doi: 10.1070/rm1998v053n02ABEH000017. Google Scholar [13] G. Keller, Equilibrium States in Ergodic Theory,, Cambridge University Press, (1998). doi: 10.1017/CBO9781107359987. Google Scholar [14] O. Lanford and D. Ruelle, Observables at infinity and states with short range correlations in statistical mechanics,, Comm. Math. Phys., 13 (1969), 194. doi: 10.1007/BF01645487. Google Scholar [15] F. Ledrappier, On Omri Sarig's work on the dynamics on surfaces,, J. Modern Dynamics, (2014). Google Scholar [16] R. Mauldin and M. Urbański, Gibbs states on the symbolic space over an infinite alphabet,, Israel J. Math., 125 (2001), 93. doi: 10.1007/BF02773377. Google Scholar [17] W. Parry, Intrinsic Markov chains,, Trans. AMS, 112 (1964), 55. doi: 10.1090/S0002-9947-1964-0161372-1. Google Scholar [18] D. Ruelle, Statistical mechanics of a one-dimensional lattice gas,, Comm. Math. Phys., 9 (1968), 267. doi: 10.1007/BF01654281. Google Scholar [19] D. Ruelle, Thermodynamic Formalism. The Mathematical Structures of Equilibrium Statistical Mechanics,, 2nd edition, (2004). doi: 10.1017/CBO9780511617546. Google Scholar [20] D. Ruelle, A variational formulation of equilibrium statistical mechanics and the Gibbs state rule,, Comm. Math. Phys., 5 (1967), 324. doi: 10.1007/BF01646446. Google Scholar [21] O. Sarig, Thermodynamic formalism for countable Markov shifts,, Ergod. Theory and Dyn. Syst., 19 (1999), 1565. doi: 10.1017/S0143385799146820. Google Scholar [22] O. Sarig, Thermodynamic formalism for null recurrent potentials,, Israel J. Math., 121 (2001), 285. doi: 10.1007/BF02802508. Google Scholar [23] O. Sarig, Phase transitions for countable Markov shifts,, Comm. Math. Phys., 217 (2001), 555. doi: 10.1007/s002200100367. Google Scholar [24] O. Sarig, On an example with a non-analytic topological pressure,, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 311. doi: 10.1016/S0764-4442(00)00189-0. Google Scholar [25] O. Sarig, Existence of Gibbs measures for countable Markov shifts,, Proc. of AMS, 131 (2003), 1751. doi: 10.1090/S0002-9939-03-06927-2. Google Scholar [26] O. Sarig, Thermodynamic Formalism for Countable Markov Shifts,, Ergodic Theory Dynam. Systems, 19 (1999), 1565. doi: 10.1017/S0143385799146820. Google Scholar [27] O. Sarig, Symbolic dynamics for surface diffeomorphisms with positive entropy,, J. Amer. Math. Soc., 26 (2013), 341. doi: 10.1090/S0894-0347-2012-00758-9. Google Scholar [28] Y. Sinai, Gibbs measures in ergodic theory,, Uspehi Mat. Nauk, 27 (1972), 21. Google Scholar [29] Y. Sinai, Construction of Markov partitions,, Functional Anal. and Appl., 2 (1968), 245. doi: 10.1007/BF01076126. Google Scholar [30] D. Vere-Jones, Geometric ergodicity in denumerable Markov chains,, Quart. J. Math. Oxford Ser. (2), 13 (1962), 7. doi: 10.1093/qmath/13.1.7. Google Scholar [31] D. Vere-Jones, Ergodic properties of nonnegative matrices. I,, Pac. J. Math., 22 (1967), 361. doi: 10.2140/pjm.1967.22.361. Google Scholar [32] P. Walter, Ruelle's operator theorem and $g$-measures,, Trans. AMS, 214 (1975), 375. Google Scholar [33] P. Walter, Invariant measures and equilibrium states for some mappings which expand distances,, Trans. AMS, 236 (1978), 121. doi: 10.1090/S0002-9947-1978-0466493-1. Google Scholar [34] P. Walter, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, (1982). Google Scholar [35] M. Yuri, Multi-dimensional maps with infinite invariant measures and countable state sofic shifts,, Indag. Math. (N. S.), 6 (1995), 355. doi: 10.1016/0019-3577(95)93202-L. Google Scholar [36] M. Yuri, On the convergence to equilibrium states for certain non-hyperbolic systems,, Ergod. Theory and Dyn. Syst., 17 (1997), 977. doi: 10.1017/S0143385797086240. Google Scholar

show all references

##### References:
 [1] J. Aaronson and M. Denker, Local limit theorems for Gibbs-Markov maps,, Stochastics Dyn., 1 (2001), 193. doi: 10.1142/S0219493701000114. Google Scholar [2] J. Aaronson, M. Denker and M. Urbański, Ergodic theory for Markov fibered systems and parabolic rational maps,, Trans. AMS, 337 (1993), 495. doi: 10.1090/S0002-9947-1993-1107025-2. Google Scholar [3] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,, Second revised edition, (2008). Google Scholar [4] J. Buzzi and O. Sarig, Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps,, Ergod. Th. and Dyn. Syst., 23 (2003), 1383. doi: 10.1017/S0143385703000087. Google Scholar [5] V. Cyr and O. Sarig, Spectral gap and transience for Ruelle operators on countable Markov shifts,, Comm. Math. Phys., 292 (2009), 637. doi: 10.1007/s00220-009-0891-4. Google Scholar [6] R. Dobrušin, Description of a random field by means of conditional probabilities and conditions for its regularity,, Teor. Veroyatnoistei i Primenenia, 13 (1968), 201. Google Scholar [7] R. Dobrušin, The problem of uniqueness of a Gibbsian random field and the problem of phase transitions,, Functional Anal. Appl., 2 (1968), 302. Google Scholar [8] M. Gordin, On the Central Limit Theorem for stationary processes,, (Russian) Doklady Akademii Nauk SSSR, 188 (1969), 739. Google Scholar [9] B. M. Gurevič, Topological entropy for denumerable Markov chains,, Dokl. Acad. Nauk SSSR, 187 (1969), 715. Google Scholar [10] B. M. Gurevič, Shift entropy and Markov measures in the space of paths of a countable graph,, Dokl. Acad. Nauk SSSR, 192 (1970), 963. Google Scholar [11] B. M. Gurevič, A variational characterization of one-dimensional countable state Gibbs random fields,, Z. Wahrsch. Verw. Gebiete, 68 (1984), 205. doi: 10.1007/BF00531778. Google Scholar [12] B. M. Gurevič and S. V. Savchenko, Thermodynamic formalism for symbolic Markov chainswith a countable number of states,, Uspehi. Mat. Nauk, 53 (1998), 3. doi: 10.1070/rm1998v053n02ABEH000017. Google Scholar [13] G. Keller, Equilibrium States in Ergodic Theory,, Cambridge University Press, (1998). doi: 10.1017/CBO9781107359987. Google Scholar [14] O. Lanford and D. Ruelle, Observables at infinity and states with short range correlations in statistical mechanics,, Comm. Math. Phys., 13 (1969), 194. doi: 10.1007/BF01645487. Google Scholar [15] F. Ledrappier, On Omri Sarig's work on the dynamics on surfaces,, J. Modern Dynamics, (2014). Google Scholar [16] R. Mauldin and M. Urbański, Gibbs states on the symbolic space over an infinite alphabet,, Israel J. Math., 125 (2001), 93. doi: 10.1007/BF02773377. Google Scholar [17] W. Parry, Intrinsic Markov chains,, Trans. AMS, 112 (1964), 55. doi: 10.1090/S0002-9947-1964-0161372-1. Google Scholar [18] D. Ruelle, Statistical mechanics of a one-dimensional lattice gas,, Comm. Math. Phys., 9 (1968), 267. doi: 10.1007/BF01654281. Google Scholar [19] D. Ruelle, Thermodynamic Formalism. The Mathematical Structures of Equilibrium Statistical Mechanics,, 2nd edition, (2004). doi: 10.1017/CBO9780511617546. Google Scholar [20] D. Ruelle, A variational formulation of equilibrium statistical mechanics and the Gibbs state rule,, Comm. Math. Phys., 5 (1967), 324. doi: 10.1007/BF01646446. Google Scholar [21] O. Sarig, Thermodynamic formalism for countable Markov shifts,, Ergod. Theory and Dyn. Syst., 19 (1999), 1565. doi: 10.1017/S0143385799146820. Google Scholar [22] O. Sarig, Thermodynamic formalism for null recurrent potentials,, Israel J. Math., 121 (2001), 285. doi: 10.1007/BF02802508. Google Scholar [23] O. Sarig, Phase transitions for countable Markov shifts,, Comm. Math. Phys., 217 (2001), 555. doi: 10.1007/s002200100367. Google Scholar [24] O. Sarig, On an example with a non-analytic topological pressure,, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 311. doi: 10.1016/S0764-4442(00)00189-0. Google Scholar [25] O. Sarig, Existence of Gibbs measures for countable Markov shifts,, Proc. of AMS, 131 (2003), 1751. doi: 10.1090/S0002-9939-03-06927-2. Google Scholar [26] O. Sarig, Thermodynamic Formalism for Countable Markov Shifts,, Ergodic Theory Dynam. Systems, 19 (1999), 1565. doi: 10.1017/S0143385799146820. Google Scholar [27] O. Sarig, Symbolic dynamics for surface diffeomorphisms with positive entropy,, J. Amer. Math. Soc., 26 (2013), 341. doi: 10.1090/S0894-0347-2012-00758-9. Google Scholar [28] Y. Sinai, Gibbs measures in ergodic theory,, Uspehi Mat. Nauk, 27 (1972), 21. Google Scholar [29] Y. Sinai, Construction of Markov partitions,, Functional Anal. and Appl., 2 (1968), 245. doi: 10.1007/BF01076126. Google Scholar [30] D. Vere-Jones, Geometric ergodicity in denumerable Markov chains,, Quart. J. Math. Oxford Ser. (2), 13 (1962), 7. doi: 10.1093/qmath/13.1.7. Google Scholar [31] D. Vere-Jones, Ergodic properties of nonnegative matrices. I,, Pac. J. Math., 22 (1967), 361. doi: 10.2140/pjm.1967.22.361. Google Scholar [32] P. Walter, Ruelle's operator theorem and $g$-measures,, Trans. AMS, 214 (1975), 375. Google Scholar [33] P. Walter, Invariant measures and equilibrium states for some mappings which expand distances,, Trans. AMS, 236 (1978), 121. doi: 10.1090/S0002-9947-1978-0466493-1. Google Scholar [34] P. Walter, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, (1982). Google Scholar [35] M. Yuri, Multi-dimensional maps with infinite invariant measures and countable state sofic shifts,, Indag. Math. (N. S.), 6 (1995), 355. doi: 10.1016/0019-3577(95)93202-L. Google Scholar [36] M. Yuri, On the convergence to equilibrium states for certain non-hyperbolic systems,, Ergod. Theory and Dyn. Syst., 17 (1997), 977. doi: 10.1017/S0143385797086240. Google Scholar
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