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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