Bernoulli equilibrium states for surface diffeomorphisms

Previous Article
On distortion in groups of homeomorphisms
JMD Home
This Issue
Next Article
A nondifferentiable essential irrational invariant curve for a $C^1$ symplectic twist map

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

Abstract
Related Papers

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.
[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.

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 (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.

[1]	Yair Daon. Bernoullicity of equilibrium measures on countable Markov shifts. Discrete and Continuous Dynamical Systems, 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]	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
[4]	Manfred Denker, Yuri Kifer, Manuel Stadlbauer. Thermodynamic formalism for random countable Markov shifts. Discrete and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 2015, 35 (1) : 593-594. doi: 10.3934/dcds.2015.35.593
[6]	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
[7]	Manfred G. Madritsch. Non-normal numbers with respect to Markov partitions. Discrete and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 2017, 37 (1) : 105-130. doi: 10.3934/dcds.2017005
[9]	Dominic Veconi. Equilibrium states of almost Anosov diffeomorphisms. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 767-780. doi: 10.3934/dcds.2020061
[10]	Wael Bahsoun, Paweł Góra. SRB measures for certain Markov processes. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 17-37. doi: 10.3934/dcds.2011.30.17
[11]	Thomas Ward, Yuki Yayama. Markov partitions reflecting the geometry of $\times2$, $\times3$. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 613-624. doi: 10.3934/dcds.2009.24.613
[12]	Alfonso Artigue. Robustly N-expansive surface diffeomorphisms. Discrete and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and Imaging, 2007, 1 (4) : 713-772. doi: 10.3934/ipi.2007.1.713
[16]	Peng Sun. Measures of intermediate entropies for skew product diffeomorphisms. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1219-1231. doi: 10.3934/dcds.2010.27.1219
[17]	Eleonora Catsigeras, Heber Enrich. SRB measures of certain almost hyperbolic diffeomorphisms with a tangency. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 177-202. doi: 10.3934/dcds.2001.7.177
[18]	Brian Marcus and Selim Tuncel. Powers of positive polynomials and codings of Markov chains onto Bernoulli shifts. Electronic Research Announcements, 1999, 5: 91-101.
[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]	John Franks, Michael Handel. Some virtually abelian subgroups of the group of analytic symplectic diffeomorphisms of a surface. Journal of Modern Dynamics, 2013, 7 (3) : 369-394. doi: 10.3934/jmd.2013.7.369

2021 Impact Factor: 0.641

Tools

Metrics

Other articles
by authors

on AIMS
Omri M. Sarig
on Google Scholar
Omri M. Sarig

[Back to Top]