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

Bernoulli equilibrium states for surface diffeomorphisms

Omri M. Sarig 1,

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.
Keywords: countable Markov partitions., surface diffeomorphisms, Bernoulli, equilibrium measures.
Mathematics Subject Classification: Primary: 37D35; Secondary: 37D2.
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.  Google Scholar

[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.  Google Scholar

[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.  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, Springer Verlag, Berlin-New York, 1975.  Google Scholar

[6]

J. Buzzi, Maximal entropy measures for piecewise affine surface homeomorphisms, Ergodic Theory Dynam. Systems, 29 (2009), 1723-1763. 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), 1383-1400.  Google Scholar

[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.  Google Scholar

[9]

B. P. Kitchens, "Symbolic Dynamics. One-Sided, Two-Sided and Countable State Markov Shifts," Universitext, Springer-Verlag, Berlin, 1998.  Google Scholar

[10]

F. Ledrappier, Propriétés ergodiques de mesures de Sinaï, Inst. Hautes Études Sci. Publ. Math. No., 59 (1984), 163-188.  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), 509-539. doi: 10.2307/1971328.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

D. Ornstein and B. Weiss, On the Bernoulli nature of systems with some hyperbolic structure, Ergodic Theory Dynam. Systems, 18 (1998), 441-456. doi: 10.1017/S0143385798100354.  Google Scholar

[18]

W. Parry, Intrinsic Markov chains, Trans. Amer. Math. Soc., 112 (1964), 55-66. doi: 10.1090/S0002-9947-1964-0161372-1.  Google Scholar

[19]

Y. Pesin, Characteristic Ljapunov exponents and smooth ergodic theory, Uspehi, Mat. Nauk, 32 (1977), 55-112, 287.  Google Scholar

[20]

M. Ratner, Anosov flows with Gibbs measures are also Bernoullian, Israel J. Math., 17 (1974), 380-391. doi: 10.1007/BF02757140.  Google Scholar

[21]

R. Ruelle, A measure associated with axiom-A attractors, Amer. J. Math., 98 (1976), 619-654. doi: 10.2307/2373810.  Google Scholar

[22]

O. M. Sarig, Thermodynamic formalism for null recurrent potentials, Israel J. Math., 121 (2001), 285-311. 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), 375-387.  Google Scholar

[25]

P. Walters, "Ergodic Theory, Introductory Lectures," Lecture Notes in Mathematics, 458, Springer-Verlag, Berlin-New York, 1975.  Google Scholar

[26]

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

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  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, Springer Verlag, Berlin-New York, 1975.  Google Scholar

[6]

J. Buzzi, Maximal entropy measures for piecewise affine surface homeomorphisms, Ergodic Theory Dynam. Systems, 29 (2009), 1723-1763. 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), 1383-1400.  Google Scholar

[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.  Google Scholar

[9]

B. P. Kitchens, "Symbolic Dynamics. One-Sided, Two-Sided and Countable State Markov Shifts," Universitext, Springer-Verlag, Berlin, 1998.  Google Scholar

[10]

F. Ledrappier, Propriétés ergodiques de mesures de Sinaï, Inst. Hautes Études Sci. Publ. Math. No., 59 (1984), 163-188.  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), 509-539. doi: 10.2307/1971328.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

D. Ornstein and B. Weiss, On the Bernoulli nature of systems with some hyperbolic structure, Ergodic Theory Dynam. Systems, 18 (1998), 441-456. doi: 10.1017/S0143385798100354.  Google Scholar

[18]

W. Parry, Intrinsic Markov chains, Trans. Amer. Math. Soc., 112 (1964), 55-66. doi: 10.1090/S0002-9947-1964-0161372-1.  Google Scholar

[19]

Y. Pesin, Characteristic Ljapunov exponents and smooth ergodic theory, Uspehi, Mat. Nauk, 32 (1977), 55-112, 287.  Google Scholar

[20]

M. Ratner, Anosov flows with Gibbs measures are also Bernoullian, Israel J. Math., 17 (1974), 380-391. doi: 10.1007/BF02757140.  Google Scholar

[21]

R. Ruelle, A measure associated with axiom-A attractors, Amer. J. Math., 98 (1976), 619-654. doi: 10.2307/2373810.  Google Scholar

[22]

O. M. Sarig, Thermodynamic formalism for null recurrent potentials, Israel J. Math., 121 (2001), 285-311. 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), 375-387.  Google Scholar

[25]

P. Walters, "Ergodic Theory, Introductory Lectures," Lecture Notes in Mathematics, 458, Springer-Verlag, Berlin-New York, 1975.  Google Scholar

[26]

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

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