September  2013, 33(9): 4003-4015. doi: 10.3934/dcds.2013.33.4003

Bernoullicity of equilibrium measures on countable Markov shifts

1. 

Courant Institute of Mathematical Sciences, New York University, 251 Mercer st. New York, NY, 10012-1185, United States

Received  July 2012 Revised  January 2013 Published  March 2013

We study equilibrium behavior for two-sided topological Markov shifts with a countable number of states. We assume the associated potential is Walters with finite first variation and that the shift is topologically transitive. We show the resulting equilibrium measure is Bernoulli up to a period.
Citation: Yair Daon. Bernoullicity of equilibrium measures on countable Markov shifts. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4003-4015. doi: 10.3934/dcds.2013.33.4003
References:
[1]

R. L. Adler, P. Shields and M. Smorodinsky, Irreducible Markov shifts, Ann. Math. Statist., 43 (1972), 1027-1029.

[2]

H. Berbee, Chains with infinite connections: Uniqueness and Markov representation, Probab. Theory Related Fields, 76 (1987), 243-253. doi: 10.1007/BF00319986.

[3]

T. Bousch, La condition de Walters, Ann. Sci. École Norm. Sup. (4), 34 (2001), 287-311. doi: 10.1016/S0012-9593(00)01062-4.

[4]

R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms," Second revised edition, Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin, 2008.

[5]

R. Bowen, Bernoulli equilibrium states for Axiom A diffeomorphisms, Math. Systems Theory, 8 (1974/75), 289-294.

[6]

J. Buzzi and O. Sarig, Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps, Ergodic Theory Dynam. Systems, 23 (2003), 1383-1400. doi: 10.1017/S0143385703000087.

[7]

Z. Coelho and A. N. Quas, Criteria for $\overline d$-continuity, Trans. Amer. Math. Soc., 350 (1998), 3257-3268. doi: 10.1090/S0002-9947-98-01923-0.

[8]

N. A. Friedman and D. S. Ornstein, On isomorphism of weak Bernoulli transformations, Advances in Math., 5 (1970), 365-394 (1970).

[9]

B. P. Kitchens, "Symbolic Dynamics. One-Sided, Two-Sided and Countable State Markov Shifts," Universitext, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-58822-8.

[10]

W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, No. 187-188 (1990), 268 pp.

[11]

M. Ratner, Anosov flows with Gibbs measures are also Bernoullian, Israel J. Math., 17 (1974), 380-391.

[12]

V. A. Rohlin, Exact endomorphism of a Lebesgue space, Magyar Tud. Akad. Mat. Fiz. Oszt. Közl., 14 (1964), 443-474.

[13]

D. Ruelle, Statistical mechanics on a compact set with Z action satisfying expansiveness and specification, Trans. AMS, 187 (1973), 237-251.

[14]

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

[15]

O. Sarig, Lecture notes on thermodynamic formalism for topological markov shifts, Notes from a class in PSU, (2009).

[16]

O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems, 19 (1999), 1565-1593. doi: 10.1017/S0143385799146820.

[17]

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

[18]

O. Sarig, Bernoulli equilibrium states for surface diffeomorphisms, J. Mod. Dyn., 5 (2011), 593-608.

[19]

J. G. Sinaĭ, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64.

[20]

P. Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc., 236 (1978), 121-153.

[21]

P. Walters, Erratum: "A necessary and sufficient condition for a two-sided continuous function to be cohomologous to a one-sided continuous function'', Dyn. Syst., 18 (2003), 271-278. doi: 10.1080/1468936031000095672.

[22]

P. Walters, Regularity conditions and Bernoulli properties of equilibrium states and $g$-measures, J. London Math. Soc. (2), 71 (2005), 379-396. doi: 10.1112/S0024610704006076.

[23]

P. Walters, A natural space of functions for the Ruelle operator theorem, Ergodic Theory Dynam. Systems, 27 (2007), 1323-1348 doi: 10.1017/S0143385707000028.

show all references

References:
[1]

R. L. Adler, P. Shields and M. Smorodinsky, Irreducible Markov shifts, Ann. Math. Statist., 43 (1972), 1027-1029.

[2]

H. Berbee, Chains with infinite connections: Uniqueness and Markov representation, Probab. Theory Related Fields, 76 (1987), 243-253. doi: 10.1007/BF00319986.

[3]

T. Bousch, La condition de Walters, Ann. Sci. École Norm. Sup. (4), 34 (2001), 287-311. doi: 10.1016/S0012-9593(00)01062-4.

[4]

R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms," Second revised edition, Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin, 2008.

[5]

R. Bowen, Bernoulli equilibrium states for Axiom A diffeomorphisms, Math. Systems Theory, 8 (1974/75), 289-294.

[6]

J. Buzzi and O. Sarig, Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps, Ergodic Theory Dynam. Systems, 23 (2003), 1383-1400. doi: 10.1017/S0143385703000087.

[7]

Z. Coelho and A. N. Quas, Criteria for $\overline d$-continuity, Trans. Amer. Math. Soc., 350 (1998), 3257-3268. doi: 10.1090/S0002-9947-98-01923-0.

[8]

N. A. Friedman and D. S. Ornstein, On isomorphism of weak Bernoulli transformations, Advances in Math., 5 (1970), 365-394 (1970).

[9]

B. P. Kitchens, "Symbolic Dynamics. One-Sided, Two-Sided and Countable State Markov Shifts," Universitext, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-58822-8.

[10]

W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, No. 187-188 (1990), 268 pp.

[11]

M. Ratner, Anosov flows with Gibbs measures are also Bernoullian, Israel J. Math., 17 (1974), 380-391.

[12]

V. A. Rohlin, Exact endomorphism of a Lebesgue space, Magyar Tud. Akad. Mat. Fiz. Oszt. Közl., 14 (1964), 443-474.

[13]

D. Ruelle, Statistical mechanics on a compact set with Z action satisfying expansiveness and specification, Trans. AMS, 187 (1973), 237-251.

[14]

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

[15]

O. Sarig, Lecture notes on thermodynamic formalism for topological markov shifts, Notes from a class in PSU, (2009).

[16]

O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems, 19 (1999), 1565-1593. doi: 10.1017/S0143385799146820.

[17]

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

[18]

O. Sarig, Bernoulli equilibrium states for surface diffeomorphisms, J. Mod. Dyn., 5 (2011), 593-608.

[19]

J. G. Sinaĭ, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64.

[20]

P. Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc., 236 (1978), 121-153.

[21]

P. Walters, Erratum: "A necessary and sufficient condition for a two-sided continuous function to be cohomologous to a one-sided continuous function'', Dyn. Syst., 18 (2003), 271-278. doi: 10.1080/1468936031000095672.

[22]

P. Walters, Regularity conditions and Bernoulli properties of equilibrium states and $g$-measures, J. London Math. Soc. (2), 71 (2005), 379-396. doi: 10.1112/S0024610704006076.

[23]

P. Walters, A natural space of functions for the Ruelle operator theorem, Ergodic Theory Dynam. Systems, 27 (2007), 1323-1348 doi: 10.1017/S0143385707000028.

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