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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

G. Keller, Equilibrium States in Ergodic Theory, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9781107359987.

[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.

[15]

F. Ledrappier, On Omri Sarig's work on the dynamics on surfaces, J. Modern Dynamics, (2014).

[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.

[17]

W. Parry, Intrinsic Markov chains, Trans. AMS, 112 (1964), 55-66. doi: 10.1090/S0002-9947-1964-0161372-1.

[18]

D. Ruelle, Statistical mechanics of a one-dimensional lattice gas, Comm. Math. Phys., 9 (1968), 267-278. doi: 10.1007/BF01654281.

[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.

[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.

[21]

O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergod. Theory and Dyn. Syst., 19 (1999), 1565-1593. doi: 10.1017/S0143385799146820.

[22]

O. Sarig, Thermodynamic formalism for null recurrent potentials, Israel J. Math., 121 (2001), 285-311. doi: 10.1007/BF02802508.

[23]

O. Sarig, Phase transitions for countable Markov shifts, Comm. Math. Phys., 217 (2001), 555-577. doi: 10.1007/s002200100367.

[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.

[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.

[26]

O. Sarig, Thermodynamic Formalism for Countable Markov Shifts, Ergodic Theory Dynam. Systems, 19 (1999), 1565-1593. doi: 10.1017/S0143385799146820.

[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.

[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.

[29]

Y. Sinai, Construction of Markov partitions, Functional Anal. and Appl., 2 (1968), 245-253. doi: 10.1007/BF01076126.

[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.

[31]

D. Vere-Jones, Ergodic properties of nonnegative matrices. I, Pac. J. Math., 22 (1967), 361-386. doi: 10.2140/pjm.1967.22.361.

[32]

P. Walter, Ruelle's operator theorem and $g$-measures, Trans. AMS, 214 (1975), 375-387.

[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.

[34]

P. Walter, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

G. Keller, Equilibrium States in Ergodic Theory, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9781107359987.

[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.

[15]

F. Ledrappier, On Omri Sarig's work on the dynamics on surfaces, J. Modern Dynamics, (2014).

[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.

[17]

W. Parry, Intrinsic Markov chains, Trans. AMS, 112 (1964), 55-66. doi: 10.1090/S0002-9947-1964-0161372-1.

[18]

D. Ruelle, Statistical mechanics of a one-dimensional lattice gas, Comm. Math. Phys., 9 (1968), 267-278. doi: 10.1007/BF01654281.

[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.

[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.

[21]

O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergod. Theory and Dyn. Syst., 19 (1999), 1565-1593. doi: 10.1017/S0143385799146820.

[22]

O. Sarig, Thermodynamic formalism for null recurrent potentials, Israel J. Math., 121 (2001), 285-311. doi: 10.1007/BF02802508.

[23]

O. Sarig, Phase transitions for countable Markov shifts, Comm. Math. Phys., 217 (2001), 555-577. doi: 10.1007/s002200100367.

[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.

[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.

[26]

O. Sarig, Thermodynamic Formalism for Countable Markov Shifts, Ergodic Theory Dynam. Systems, 19 (1999), 1565-1593. doi: 10.1017/S0143385799146820.

[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.

[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.

[29]

Y. Sinai, Construction of Markov partitions, Functional Anal. and Appl., 2 (1968), 245-253. doi: 10.1007/BF01076126.

[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.

[31]

D. Vere-Jones, Ergodic properties of nonnegative matrices. I, Pac. J. Math., 22 (1967), 361-386. doi: 10.2140/pjm.1967.22.361.

[32]

P. Walter, Ruelle's operator theorem and $g$-measures, Trans. AMS, 214 (1975), 375-387.

[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.

[34]

P. Walter, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

[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.

[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.

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