American Institute of Mathematical Sciences

July  2011, 5(3): 593-608. doi: 10.3934/jmd.2011.5.593

Bernoulli equilibrium states for surface diffeomorphisms

 1 Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, POB 26, Rehovot, Israel

Received  May 2011 Revised  July 2011 Published  November 2011

Suppose $f\colon M\to M$ is a $C^{1+\alpha}$ $(\alpha>0)$ diffeomorphism on a compact smooth orientable manifold $M$ of dimension 2, and let $\mu_\Psi$ be an equilibrium measure for a Hölder-continuous potential $\Psi\colon M\to \mathbb R$. We show that if $\mu_\Psi$ has positive measure-theoretic entropy, then $f$ is measure-theoretically isomorphic mod $\mu_\Psi$ to the product of a Bernoulli scheme and a finite rotation.
Citation: Omri M. Sarig. Bernoulli equilibrium states for surface diffeomorphisms. Journal of Modern Dynamics, 2011, 5 (3) : 593-608. doi: 10.3934/jmd.2011.5.593
References:
 [1] R. L. Adler and B. Weiss, "Similarity of Automorphisms of the Torus," Memoirs of the American Mathematical Society, No. 98, American Mathematical Society, Providence, R.I., 1970. [2] R. L. Adler, P. Shields and M. Smorodinsky, Irreducible Markov shifts, The Annals of Math. Statistics, 43 (1972), 1027-1029. doi: 10.1214/aoms/1177692569. [3] L. Barreira and Y. Pesin, "Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents," Encyclopedia of Mathematics and its Applications, 115, Cambridge University Press, Cambridge, 2007. [4] R. Bowen, Bernoulli equilibrium states for Axiom A diffeomorphisms, Math. Systems Theory, 8 (1974/75), 289-294. doi: 10.1007/BF01780576. [5] R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms," Lecture Notes in Mathematics, 470, Springer Verlag, Berlin-New York, 1975. [6] J. Buzzi, Maximal entropy measures for piecewise affine surface homeomorphisms, Ergodic Theory Dynam. Systems, 29 (2009), 1723-1763. doi: 10.1017/S0143385708000953. [7] J. Buzzi and O. Sarig, Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps, Ergodic Th. & Dynam. Syst., 23 (2003), 1383-1400. [8] B. M. Gurevič, Shift entropy and Markov measures in the space of paths of a countable graph, (Russian), Dokl. Akad. Nauk SSSR, 192 (1970), 963-965; English Transl. in Soviet Math. Dokl., 11 (1970), 744-747. [9] B. P. Kitchens, "Symbolic Dynamics. One-Sided, Two-Sided and Countable State Markov Shifts," Universitext, Springer-Verlag, Berlin, 1998. [10] F. Ledrappier, Propriétés ergodiques de mesures de Sinaï, Inst. Hautes Études Sci. Publ. Math. No., 59 (1984), 163-188. [11] F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin's entropy formula, Ann. of Math. (2), 122 (1985), 509-539. doi: 10.2307/1971328. [12] S. Newhouse, Continuity properties of entropy, Annals of Math. (2), 129 (1989), 215-235; Errata in Annals of Math., 131 (1990), 409-410. doi: 10.2307/1971492. [13] D. Ornstein, Factors of Bernoulli shifts are Bernoulli shifts, Adv. in Math., 5 (1970), 349-364. doi: 10.1016/0001-8708(70)90009-5. [14] D. Ornstein, Two Bernoulli shifts with infinite entropy are isomorphic, Adv. in Math., 5 (1970), 339-348. doi: 10.1016/0001-8708(70)90008-3. [15] D. Ornstein, Imbedding Bernoulli shifts in flows, in "1970 Contributions to Ergodic Theory and Probability" (Proc. Conf., Ohio State Univ., Columbus, Ohio, 1970), 178-218, Springer, Berlin, 1970. [16] D. Ornstein and N. A. Friedman, On isomorphism of weak Bernoulli transformations, Adv. in Math., 5 (1970), 365-394. doi: 10.1016/0001-8708(70)90010-1. [17] D. Ornstein and B. Weiss, On the Bernoulli nature of systems with some hyperbolic structure, Ergodic Theory Dynam. Systems, 18 (1998), 441-456. doi: 10.1017/S0143385798100354. [18] W. Parry, Intrinsic Markov chains, Trans. Amer. Math. Soc., 112 (1964), 55-66. doi: 10.1090/S0002-9947-1964-0161372-1. [19] Y. Pesin, Characteristic Ljapunov exponents and smooth ergodic theory, Uspehi, Mat. Nauk, 32 (1977), 55-112, 287. [20] M. Ratner, Anosov flows with Gibbs measures are also Bernoullian, Israel J. Math., 17 (1974), 380-391. doi: 10.1007/BF02757140. [21] R. Ruelle, A measure associated with axiom-A attractors, Amer. J. Math., 98 (1976), 619-654. doi: 10.2307/2373810. [22] O. M. Sarig, Thermodynamic formalism for null recurrent potentials, Israel J. Math., 121 (2001), 285-311. doi: 10.1007/BF02802508. [23] O. M. Sarig, Symbolic dynamics for surface diffeomorphisms with positive entropy, submitted. [24] P. Walters, Ruelle's operator theorem and g-measures, Trans. Amer. Math. Soc., 214 (1975), 375-387. [25] P. Walters, "Ergodic Theory, Introductory Lectures," Lecture Notes in Mathematics, 458, Springer-Verlag, Berlin-New York, 1975. [26] 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.

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References:
 [1] R. L. Adler and B. Weiss, "Similarity of Automorphisms of the Torus," Memoirs of the American Mathematical Society, No. 98, American Mathematical Society, Providence, R.I., 1970. [2] R. L. Adler, P. Shields and M. Smorodinsky, Irreducible Markov shifts, The Annals of Math. Statistics, 43 (1972), 1027-1029. doi: 10.1214/aoms/1177692569. [3] L. Barreira and Y. Pesin, "Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents," Encyclopedia of Mathematics and its Applications, 115, Cambridge University Press, Cambridge, 2007. [4] R. Bowen, Bernoulli equilibrium states for Axiom A diffeomorphisms, Math. Systems Theory, 8 (1974/75), 289-294. doi: 10.1007/BF01780576. [5] R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms," Lecture Notes in Mathematics, 470, Springer Verlag, Berlin-New York, 1975. [6] J. Buzzi, Maximal entropy measures for piecewise affine surface homeomorphisms, Ergodic Theory Dynam. Systems, 29 (2009), 1723-1763. doi: 10.1017/S0143385708000953. [7] J. Buzzi and O. Sarig, Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps, Ergodic Th. & Dynam. Syst., 23 (2003), 1383-1400. [8] B. M. Gurevič, Shift entropy and Markov measures in the space of paths of a countable graph, (Russian), Dokl. Akad. Nauk SSSR, 192 (1970), 963-965; English Transl. in Soviet Math. Dokl., 11 (1970), 744-747. [9] B. P. Kitchens, "Symbolic Dynamics. One-Sided, Two-Sided and Countable State Markov Shifts," Universitext, Springer-Verlag, Berlin, 1998. [10] F. Ledrappier, Propriétés ergodiques de mesures de Sinaï, Inst. Hautes Études Sci. Publ. Math. No., 59 (1984), 163-188. [11] F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin's entropy formula, Ann. of Math. (2), 122 (1985), 509-539. doi: 10.2307/1971328. [12] S. Newhouse, Continuity properties of entropy, Annals of Math. (2), 129 (1989), 215-235; Errata in Annals of Math., 131 (1990), 409-410. doi: 10.2307/1971492. [13] D. Ornstein, Factors of Bernoulli shifts are Bernoulli shifts, Adv. in Math., 5 (1970), 349-364. doi: 10.1016/0001-8708(70)90009-5. [14] D. Ornstein, Two Bernoulli shifts with infinite entropy are isomorphic, Adv. in Math., 5 (1970), 339-348. doi: 10.1016/0001-8708(70)90008-3. [15] D. Ornstein, Imbedding Bernoulli shifts in flows, in "1970 Contributions to Ergodic Theory and Probability" (Proc. Conf., Ohio State Univ., Columbus, Ohio, 1970), 178-218, Springer, Berlin, 1970. [16] D. Ornstein and N. A. Friedman, On isomorphism of weak Bernoulli transformations, Adv. in Math., 5 (1970), 365-394. doi: 10.1016/0001-8708(70)90010-1. [17] D. Ornstein and B. Weiss, On the Bernoulli nature of systems with some hyperbolic structure, Ergodic Theory Dynam. Systems, 18 (1998), 441-456. doi: 10.1017/S0143385798100354. [18] W. Parry, Intrinsic Markov chains, Trans. Amer. Math. Soc., 112 (1964), 55-66. doi: 10.1090/S0002-9947-1964-0161372-1. [19] Y. Pesin, Characteristic Ljapunov exponents and smooth ergodic theory, Uspehi, Mat. Nauk, 32 (1977), 55-112, 287. [20] M. Ratner, Anosov flows with Gibbs measures are also Bernoullian, Israel J. Math., 17 (1974), 380-391. doi: 10.1007/BF02757140. [21] R. Ruelle, A measure associated with axiom-A attractors, Amer. J. Math., 98 (1976), 619-654. doi: 10.2307/2373810. [22] O. M. Sarig, Thermodynamic formalism for null recurrent potentials, Israel J. Math., 121 (2001), 285-311. doi: 10.1007/BF02802508. [23] O. M. Sarig, Symbolic dynamics for surface diffeomorphisms with positive entropy, submitted. [24] P. Walters, Ruelle's operator theorem and g-measures, Trans. Amer. Math. Soc., 214 (1975), 375-387. [25] P. Walters, "Ergodic Theory, Introductory Lectures," Lecture Notes in Mathematics, 458, Springer-Verlag, Berlin-New York, 1975. [26] 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.
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