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 (): 289.  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.

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.

[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 (): 289.  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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