# 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 (Brin Prize article)

 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'sprofound contributions to statistical physics and in particular,thermodynamic formalism for countable Markov shifts. I will discusssome of Sarig's work on characterization of existence of Gibbsmeasures, existence and uniqueness of equilibrium states as well asphase 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), 193-237. 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), 495-548. 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, Edited by J.-R. Chazottes, Lect. Notes Math., 470, Springer-Verlag, Berlin, 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), 1383-1400. 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), 637-666. 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), 201-229; English translation in Theory of Prob. and Appl., 13 (1968), 197-223.  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), 302-312. Google Scholar [8] M. Gordin, On the Central Limit Theorem for stationary processes, (Russian) Doklady Akademii Nauk SSSR, 188 (1969), 739-741; English translation in Soviet Math. Dokl., 10 (1969), 1174-1176.  Google Scholar [9] B. M. Gurevič, Topological entropy for denumerable Markov chains, Dokl. Acad. Nauk SSSR, 187 (1969), 715-718; English translation in Soviet Math. Dokl., 10 (1969), 911-915.  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), 963-965; English translation in Soviet Math. Dokl., 11 (1970), 744-747.  Google Scholar [11] B. M. Gurevič, A variational characterization of one-dimensional countable state Gibbs random fields, Z. Wahrsch. Verw. Gebiete, 68 (1984), 205-242. 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), 3-106; English translation in Russian Math. Surv., 53 (1998), 245-344. doi: 10.1070/rm1998v053n02ABEH000017.  Google Scholar [13] G. Keller, Equilibrium States in Ergodic Theory, Cambridge University Press, Cambridge, 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), 194-215. 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), 93-130. doi: 10.1007/BF02773377.  Google Scholar [17] W. Parry, Intrinsic Markov chains, Trans. AMS, 112 (1964), 55-66. 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), 267-278. doi: 10.1007/BF01654281.  Google Scholar [19] D. Ruelle, Thermodynamic Formalism. The Mathematical Structures of Equilibrium Statistical Mechanics, 2nd edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 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), 324-329. doi: 10.1007/BF01646446.  Google Scholar [21] O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergod. Theory and Dyn. Syst., 19 (1999), 1565-1593. doi: 10.1017/S0143385799146820.  Google Scholar [22] O. Sarig, Thermodynamic formalism for null recurrent potentials, Israel J. Math., 121 (2001), 285-311. doi: 10.1007/BF02802508.  Google Scholar [23] O. Sarig, Phase transitions for countable Markov shifts, Comm. Math. Phys., 217 (2001), 555-577. 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), 311-315. 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), 1751-1758. 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), 1565-1593. doi: 10.1017/S0143385799146820.  Google Scholar [27] O. Sarig, Symbolic dynamics for surface diffeomorphisms with positive entropy, J. Amer. Math. Soc., 26 (2013), 341-426. doi: 10.1090/S0894-0347-2012-00758-9.  Google Scholar [28] Y. Sinai, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64; English translation in Russan Math. Surveys, 27 (1972), 21-69.  Google Scholar [29] Y. Sinai, Construction of Markov partitions, Functional Anal. and Appl., 2 (1968), 245-253. doi: 10.1007/BF01076126.  Google Scholar [30] D. Vere-Jones, Geometric ergodicity in denumerable Markov chains, Quart. J. Math. Oxford Ser. (2), 13 (1962), 7-28. doi: 10.1093/qmath/13.1.7.  Google Scholar [31] D. Vere-Jones, Ergodic properties of nonnegative matrices. I, Pac. J. Math., 22 (1967), 361-386. doi: 10.2140/pjm.1967.22.361.  Google Scholar [32] P. Walter, Ruelle's operator theorem and $g$-measures, Trans. AMS, 214 (1975), 375-387.  Google Scholar [33] P. Walter, Invariant measures and equilibrium states for some mappings which expand distances, Trans. AMS, 236 (1978), 121-153. doi: 10.1090/S0002-9947-1978-0466493-1.  Google Scholar [34] P. Walter, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar [35] M. Yuri, Multi-dimensional maps with infinite invariant measures and countable state sofic shifts, Indag. Math. (N. S.), 6 (1995), 355-383. 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), 977-1000. 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), 193-237. 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), 495-548. 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, Edited by J.-R. Chazottes, Lect. Notes Math., 470, Springer-Verlag, Berlin, 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), 1383-1400. 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), 637-666. 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), 201-229; English translation in Theory of Prob. and Appl., 13 (1968), 197-223.  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), 302-312. Google Scholar [8] M. Gordin, On the Central Limit Theorem for stationary processes, (Russian) Doklady Akademii Nauk SSSR, 188 (1969), 739-741; English translation in Soviet Math. Dokl., 10 (1969), 1174-1176.  Google Scholar [9] B. M. Gurevič, Topological entropy for denumerable Markov chains, Dokl. Acad. Nauk SSSR, 187 (1969), 715-718; English translation in Soviet Math. Dokl., 10 (1969), 911-915.  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), 963-965; English translation in Soviet Math. Dokl., 11 (1970), 744-747.  Google Scholar [11] B. M. Gurevič, A variational characterization of one-dimensional countable state Gibbs random fields, Z. Wahrsch. Verw. Gebiete, 68 (1984), 205-242. 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), 3-106; English translation in Russian Math. Surv., 53 (1998), 245-344. doi: 10.1070/rm1998v053n02ABEH000017.  Google Scholar [13] G. Keller, Equilibrium States in Ergodic Theory, Cambridge University Press, Cambridge, 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), 194-215. 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), 93-130. doi: 10.1007/BF02773377.  Google Scholar [17] W. Parry, Intrinsic Markov chains, Trans. AMS, 112 (1964), 55-66. 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), 267-278. doi: 10.1007/BF01654281.  Google Scholar [19] D. Ruelle, Thermodynamic Formalism. The Mathematical Structures of Equilibrium Statistical Mechanics, 2nd edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 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), 324-329. doi: 10.1007/BF01646446.  Google Scholar [21] O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergod. Theory and Dyn. Syst., 19 (1999), 1565-1593. doi: 10.1017/S0143385799146820.  Google Scholar [22] O. Sarig, Thermodynamic formalism for null recurrent potentials, Israel J. Math., 121 (2001), 285-311. doi: 10.1007/BF02802508.  Google Scholar [23] O. Sarig, Phase transitions for countable Markov shifts, Comm. Math. Phys., 217 (2001), 555-577. 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), 311-315. 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), 1751-1758. 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), 1565-1593. doi: 10.1017/S0143385799146820.  Google Scholar [27] O. Sarig, Symbolic dynamics for surface diffeomorphisms with positive entropy, J. Amer. Math. Soc., 26 (2013), 341-426. doi: 10.1090/S0894-0347-2012-00758-9.  Google Scholar [28] Y. Sinai, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64; English translation in Russan Math. Surveys, 27 (1972), 21-69.  Google Scholar [29] Y. Sinai, Construction of Markov partitions, Functional Anal. and Appl., 2 (1968), 245-253. doi: 10.1007/BF01076126.  Google Scholar [30] D. Vere-Jones, Geometric ergodicity in denumerable Markov chains, Quart. J. Math. Oxford Ser. (2), 13 (1962), 7-28. doi: 10.1093/qmath/13.1.7.  Google Scholar [31] D. Vere-Jones, Ergodic properties of nonnegative matrices. I, Pac. J. Math., 22 (1967), 361-386. doi: 10.2140/pjm.1967.22.361.  Google Scholar [32] P. Walter, Ruelle's operator theorem and $g$-measures, Trans. AMS, 214 (1975), 375-387.  Google Scholar [33] P. Walter, Invariant measures and equilibrium states for some mappings which expand distances, Trans. AMS, 236 (1978), 121-153. doi: 10.1090/S0002-9947-1978-0466493-1.  Google Scholar [34] P. Walter, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar [35] M. Yuri, Multi-dimensional maps with infinite invariant measures and countable state sofic shifts, Indag. Math. (N. S.), 6 (1995), 355-383. 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), 977-1000. doi: 10.1017/S0143385797086240.  Google Scholar
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