# American Institute of Mathematical Sciences

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 & 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.  Google Scholar [2] H. Berbee, Chains with infinite connections: Uniqueness and Markov representation, Probab. Theory Related Fields, 76 (1987), 243-253. doi: 10.1007/BF00319986.  Google Scholar [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.  Google Scholar [4] R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms," Second revised edition, Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin, 2008.  Google Scholar [5] R. Bowen, Bernoulli equilibrium states for Axiom A diffeomorphisms,, Math. Systems Theory, 8 (): 289.   Google Scholar [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.  Google Scholar [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.  Google Scholar [8] N. A. Friedman and D. S. Ornstein, On isomorphism of weak Bernoulli transformations, Advances in Math., 5 (1970), 365-394 (1970).  Google Scholar [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.  Google Scholar [10] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, No. 187-188 (1990), 268 pp.  Google Scholar [11] M. Ratner, Anosov flows with Gibbs measures are also Bernoullian, Israel J. Math., 17 (1974), 380-391.  Google Scholar [12] V. A. Rohlin, Exact endomorphism of a Lebesgue space, Magyar Tud. Akad. Mat. Fiz. Oszt. Közl., 14 (1964), 443-474.  Google Scholar [13] D. Ruelle, Statistical mechanics on a compact set with Z action satisfying expansiveness and specification, Trans. AMS, 187 (1973), 237-251.  Google Scholar [14] D. Ruelle, Statistical mechanics of a one-dimensional lattice gas, Comm. Math. Phys., 9 (1968), 267-278.  Google Scholar [15] O. Sarig, Lecture notes on thermodynamic formalism for topological markov shifts, Notes from a class in PSU, (2009). Google Scholar [16] O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems, 19 (1999), 1565-1593. doi: 10.1017/S0143385799146820.  Google Scholar [17] O. Sarig, Thermodynamic formalism for null recurrent potentials, Israel J. Math., 121 (2001), 285-311. doi: 10.1007/BF02802508.  Google Scholar [18] O. Sarig, Bernoulli equilibrium states for surface diffeomorphisms, J. Mod. Dyn., 5 (2011), 593-608.  Google Scholar [19] J. G. Sinaĭ, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64.  Google Scholar [20] P. Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc., 236 (1978), 121-153.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### References:
 [1] R. L. Adler, P. Shields and M. Smorodinsky, Irreducible Markov shifts, Ann. Math. Statist., 43 (1972), 1027-1029.  Google Scholar [2] H. Berbee, Chains with infinite connections: Uniqueness and Markov representation, Probab. Theory Related Fields, 76 (1987), 243-253. doi: 10.1007/BF00319986.  Google Scholar [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.  Google Scholar [4] R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms," Second revised edition, Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin, 2008.  Google Scholar [5] R. Bowen, Bernoulli equilibrium states for Axiom A diffeomorphisms,, Math. Systems Theory, 8 (): 289.   Google Scholar [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.  Google Scholar [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.  Google Scholar [8] N. A. Friedman and D. S. Ornstein, On isomorphism of weak Bernoulli transformations, Advances in Math., 5 (1970), 365-394 (1970).  Google Scholar [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.  Google Scholar [10] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, No. 187-188 (1990), 268 pp.  Google Scholar [11] M. Ratner, Anosov flows with Gibbs measures are also Bernoullian, Israel J. Math., 17 (1974), 380-391.  Google Scholar [12] V. A. Rohlin, Exact endomorphism of a Lebesgue space, Magyar Tud. Akad. Mat. Fiz. Oszt. Közl., 14 (1964), 443-474.  Google Scholar [13] D. Ruelle, Statistical mechanics on a compact set with Z action satisfying expansiveness and specification, Trans. AMS, 187 (1973), 237-251.  Google Scholar [14] D. Ruelle, Statistical mechanics of a one-dimensional lattice gas, Comm. Math. Phys., 9 (1968), 267-278.  Google Scholar [15] O. Sarig, Lecture notes on thermodynamic formalism for topological markov shifts, Notes from a class in PSU, (2009). Google Scholar [16] O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems, 19 (1999), 1565-1593. doi: 10.1017/S0143385799146820.  Google Scholar [17] O. Sarig, Thermodynamic formalism for null recurrent potentials, Israel J. Math., 121 (2001), 285-311. doi: 10.1007/BF02802508.  Google Scholar [18] O. Sarig, Bernoulli equilibrium states for surface diffeomorphisms, J. Mod. Dyn., 5 (2011), 593-608.  Google Scholar [19] J. G. Sinaĭ, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64.  Google Scholar [20] P. Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc., 236 (1978), 121-153.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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