# 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, (1970). Google Scholar [2] R. L. Adler, P. Shields and M. Smorodinsky, Irreducible Markov shifts,, The Annals of Math. Statistics, 43 (1972), 1027. doi: 10.1214/aoms/1177692569. Google Scholar [3] L. Barreira and Y. Pesin, "Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents,", Encyclopedia of Mathematics and its Applications, 115 (2007). Google Scholar [4] R. Bowen, Bernoulli equilibrium states for Axiom A diffeomorphisms,, Math. Systems Theory, 8 (): 289. doi: 10.1007/BF01780576. Google Scholar [5] R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,", Lecture Notes in Mathematics, 470 (1975). Google Scholar [6] J. Buzzi, Maximal entropy measures for piecewise affine surface homeomorphisms,, Ergodic Theory Dynam. Systems, 29 (2009), 1723. doi: 10.1017/S0143385708000953. Google Scholar [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. Google Scholar [8] B. M. Gurevič, Shift entropy and Markov measures in the space of paths of a countable graph,, (Russian), 192 (1970), 963. Google Scholar [9] B. P. Kitchens, "Symbolic Dynamics. One-Sided, Two-Sided and Countable State Markov Shifts,", Universitext, (1998). Google Scholar [10] F. Ledrappier, Propriétés ergodiques de mesures de Sinaï,, Inst. Hautes Études Sci. Publ. Math. No., 59 (1984), 163. Google Scholar [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. doi: 10.2307/1971328. Google Scholar [12] S. Newhouse, Continuity properties of entropy,, Annals of Math. (2), 129 (1989), 215. doi: 10.2307/1971492. Google Scholar [13] D. Ornstein, Factors of Bernoulli shifts are Bernoulli shifts,, Adv. in Math., 5 (1970), 349. doi: 10.1016/0001-8708(70)90009-5. Google Scholar [14] D. Ornstein, Two Bernoulli shifts with infinite entropy are isomorphic,, Adv. in Math., 5 (1970), 339. doi: 10.1016/0001-8708(70)90008-3. Google Scholar [15] D. Ornstein, Imbedding Bernoulli shifts in flows,, in, (1970), 178. Google Scholar [16] D. Ornstein and N. A. Friedman, On isomorphism of weak Bernoulli transformations,, Adv. in Math., 5 (1970), 365. doi: 10.1016/0001-8708(70)90010-1. Google Scholar [17] D. Ornstein and B. Weiss, On the Bernoulli nature of systems with some hyperbolic structure,, Ergodic Theory Dynam. Systems, 18 (1998), 441. doi: 10.1017/S0143385798100354. Google Scholar [18] W. Parry, Intrinsic Markov chains,, Trans. Amer. Math. Soc., 112 (1964), 55. doi: 10.1090/S0002-9947-1964-0161372-1. Google Scholar [19] Y. Pesin, Characteristic Ljapunov exponents and smooth ergodic theory,, Uspehi, 32 (1977), 55. Google Scholar [20] M. Ratner, Anosov flows with Gibbs measures are also Bernoullian,, Israel J. Math., 17 (1974), 380. doi: 10.1007/BF02757140. Google Scholar [21] R. Ruelle, A measure associated with axiom-A attractors,, Amer. J. Math., 98 (1976), 619. doi: 10.2307/2373810. Google Scholar [22] O. M. Sarig, Thermodynamic formalism for null recurrent potentials,, Israel J. Math., 121 (2001), 285. doi: 10.1007/BF02802508. Google Scholar [23] O. M. Sarig, Symbolic dynamics for surface diffeomorphisms with positive entropy,, submitted., (). Google Scholar [24] P. Walters, Ruelle's operator theorem and g-measures,, Trans. Amer. Math. Soc., 214 (1975), 375. Google Scholar [25] P. Walters, "Ergodic Theory, Introductory Lectures,", Lecture Notes in Mathematics, 458 (1975). Google Scholar [26] P. Walters, Regularity conditions and Bernoulli properties of equilibrium states and g-measures,, J. London Math. Soc. (2), 71 (2005), 379. doi: 10.1112/S0024610704006076. Google Scholar

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##### References:
 [1] R. L. Adler and B. Weiss, "Similarity of Automorphisms of the Torus,", Memoirs of the American Mathematical Society, (1970). Google Scholar [2] R. L. Adler, P. Shields and M. Smorodinsky, Irreducible Markov shifts,, The Annals of Math. Statistics, 43 (1972), 1027. doi: 10.1214/aoms/1177692569. Google Scholar [3] L. Barreira and Y. Pesin, "Nonuniform Hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents,", Encyclopedia of Mathematics and its Applications, 115 (2007). Google Scholar [4] R. Bowen, Bernoulli equilibrium states for Axiom A diffeomorphisms,, Math. Systems Theory, 8 (): 289. doi: 10.1007/BF01780576. Google Scholar [5] R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms,", Lecture Notes in Mathematics, 470 (1975). Google Scholar [6] J. Buzzi, Maximal entropy measures for piecewise affine surface homeomorphisms,, Ergodic Theory Dynam. Systems, 29 (2009), 1723. doi: 10.1017/S0143385708000953. Google Scholar [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. Google Scholar [8] B. M. Gurevič, Shift entropy and Markov measures in the space of paths of a countable graph,, (Russian), 192 (1970), 963. Google Scholar [9] B. P. Kitchens, "Symbolic Dynamics. One-Sided, Two-Sided and Countable State Markov Shifts,", Universitext, (1998). Google Scholar [10] F. Ledrappier, Propriétés ergodiques de mesures de Sinaï,, Inst. Hautes Études Sci. Publ. Math. No., 59 (1984), 163. Google Scholar [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. doi: 10.2307/1971328. Google Scholar [12] S. Newhouse, Continuity properties of entropy,, Annals of Math. (2), 129 (1989), 215. doi: 10.2307/1971492. Google Scholar [13] D. Ornstein, Factors of Bernoulli shifts are Bernoulli shifts,, Adv. in Math., 5 (1970), 349. doi: 10.1016/0001-8708(70)90009-5. Google Scholar [14] D. Ornstein, Two Bernoulli shifts with infinite entropy are isomorphic,, Adv. in Math., 5 (1970), 339. doi: 10.1016/0001-8708(70)90008-3. Google Scholar [15] D. Ornstein, Imbedding Bernoulli shifts in flows,, in, (1970), 178. Google Scholar [16] D. Ornstein and N. A. Friedman, On isomorphism of weak Bernoulli transformations,, Adv. in Math., 5 (1970), 365. doi: 10.1016/0001-8708(70)90010-1. Google Scholar [17] D. Ornstein and B. Weiss, On the Bernoulli nature of systems with some hyperbolic structure,, Ergodic Theory Dynam. Systems, 18 (1998), 441. doi: 10.1017/S0143385798100354. Google Scholar [18] W. Parry, Intrinsic Markov chains,, Trans. Amer. Math. Soc., 112 (1964), 55. doi: 10.1090/S0002-9947-1964-0161372-1. Google Scholar [19] Y. Pesin, Characteristic Ljapunov exponents and smooth ergodic theory,, Uspehi, 32 (1977), 55. Google Scholar [20] M. Ratner, Anosov flows with Gibbs measures are also Bernoullian,, Israel J. Math., 17 (1974), 380. doi: 10.1007/BF02757140. Google Scholar [21] R. Ruelle, A measure associated with axiom-A attractors,, Amer. J. Math., 98 (1976), 619. doi: 10.2307/2373810. Google Scholar [22] O. M. Sarig, Thermodynamic formalism for null recurrent potentials,, Israel J. Math., 121 (2001), 285. doi: 10.1007/BF02802508. Google Scholar [23] O. M. Sarig, Symbolic dynamics for surface diffeomorphisms with positive entropy,, submitted., (). Google Scholar [24] P. Walters, Ruelle's operator theorem and g-measures,, Trans. Amer. Math. Soc., 214 (1975), 375. Google Scholar [25] P. Walters, "Ergodic Theory, Introductory Lectures,", Lecture Notes in Mathematics, 458 (1975). Google Scholar [26] P. Walters, Regularity conditions and Bernoulli properties of equilibrium states and g-measures,, J. London Math. Soc. (2), 71 (2005), 379. doi: 10.1112/S0024610704006076. Google Scholar
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