American Institute of Mathematical Sciences

Advanced
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, (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

show all references

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

[1]

Yair Daon. Bernoullicity of equilibrium measures on countable Markov shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4003-4015. doi: 10.3934/dcds.2013.33.4003
[2]

Michael Jakobson, Lucia D. Simonelli. Countable Markov partitions suitable for thermodynamic formalism. Journal of Modern Dynamics, 2018, 13: 199-219. doi: 10.3934/jmd.2018018
[3]

Yakov Pesin. On the work of Sarig on countable Markov chains and thermodynamic formalism. Journal of Modern Dynamics, 2014, 8 (1) : 1-14. doi: 10.3934/jmd.2014.8.1
[4]

Manfred Denker, Yuri Kifer, Manuel Stadlbauer. Thermodynamic formalism for random countable Markov shifts. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 131-164. doi: 10.3934/dcds.2008.22.131
[5]

Manfred Denker, Yuri Kifer, Manuel Stadlbauer. Corrigendum to: Thermodynamic formalism for random countable Markov shifts. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 593-594. doi: 10.3934/dcds.2015.35.593
[6]

Enrique R. Pujals, Federico Rodriguez Hertz. Critical points for surface diffeomorphisms. Journal of Modern Dynamics, 2007, 1 (4) : 615-648. doi: 10.3934/jmd.2007.1.615
[7]

Manfred G. Madritsch. Non-normal numbers with respect to Markov partitions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 663-676. doi: 10.3934/dcds.2014.34.663
[8]

Mark F. Demers, Christopher J. Ianzano, Philip Mayer, Peter Morfe, Elizabeth C. Yoo. Limiting distributions for countable state topological Markov chains with holes. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 105-130. doi: 10.3934/dcds.2017005
[9]

Dominic Veconi. Equilibrium states of almost Anosov diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 767-780. doi: 10.3934/dcds.2020061
[10]

Thomas Ward, Yuki Yayama. Markov partitions reflecting the geometry of $\times2$, $\times3$. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 613-624. doi: 10.3934/dcds.2009.24.613
[11]

Wael Bahsoun, Paweł Góra. SRB measures for certain Markov processes. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 17-37. doi: 10.3934/dcds.2011.30.17
[12]

Alfonso Artigue. Robustly N-expansive surface diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2367-2376. doi: 10.3934/dcds.2016.36.2367
[13]

C. Morales. On spiral periodic points and saddles for surface diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1191-1195. doi: 10.3934/dcds.2011.29.1191
[14]

Yakov Pesin, Samuel Senti. Equilibrium measures for maps with inducing schemes. Journal of Modern Dynamics, 2008, 2 (3) : 397-430. doi: 10.3934/jmd.2008.2.397
[15]

Barry Simon. Equilibrium measures and capacities in spectral theory. Inverse Problems & Imaging, 2007, 1 (4) : 713-772. doi: 10.3934/ipi.2007.1.713
[16]

Brian Marcus and Selim Tuncel. Powers of positive polynomials and codings of Markov chains onto Bernoulli shifts. Electronic Research Announcements, 1999, 5: 91-101.

[17]

Peng Sun. Measures of intermediate entropies for skew product diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1219-1231. doi: 10.3934/dcds.2010.27.1219
[18]

Eleonora Catsigeras, Heber Enrich. SRB measures of certain almost hyperbolic diffeomorphisms with a tangency. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 177-202. doi: 10.3934/dcds.2001.7.177
[19]

Rostyslav Kravchenko. The action of finite-state tree automorphisms on Bernoulli measures. Journal of Modern Dynamics, 2010, 4 (3) : 443-451. doi: 10.3934/jmd.2010.4.443
[20]

Eugen Mihailescu. Equilibrium measures, prehistories distributions and fractal dimensions for endomorphisms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2485-2502. doi: 10.3934/dcds.2012.32.2485

2018 Impact Factor: 0.295

Tools

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences