Duke Mathematical Journal

New criteria for ergodicity and nonuniform hyperbolicity

F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi, and R. Ures

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Abstract

In this work we obtain a new criterion to establish ergodicity and nonuniform hyperbolicity of smooth measures of diffeomorphisms of closed connected Riemannian manifolds. This method allows us to give a more accurate description of certain ergodic components. The use of this criterion in combination with topological devices such as blenders lets us obtain global ergodicity and abundance of nonzero Lyapunov exponents in some contexts.

In the partial hyperbolicity context, we obtain that stably ergodic diffeomorphisms are C1-dense among volume-preserving partially hyperbolic diffeomorphisms with 2-dimensional center bundle. This is motivated by a well-known conjecture of Pugh and Shub.

Article information

Source
Duke Math. J., Volume 160, Number 3 (2011), 599-629.

Dates
First available in Project Euclid: 7 November 2011

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1320674938

Digital Object Identifier
doi:10.1215/00127094-1444314

Mathematical Reviews number (MathSciNet)
MR2852370

Zentralblatt MATH identifier
1290.37011

Subjects
Primary: 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
Secondary: 37D30: Partially hyperbolic systems and dominated splittings 37D35: Thermodynamic formalism, variational principles, equilibrium states

Citation

Rodriguez Hertz, F.; Rodriguez Hertz, M. A.; Tahzibi, A.; Ures, R. New criteria for ergodicity and nonuniform hyperbolicity. Duke Math. J. 160 (2011), no. 3, 599--629. doi:10.1215/00127094-1444314. https://projecteuclid.org/euclid.dmj/1320674938


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References

  • [1] D. V. Anosov, Geodesic Flows on Closed Riemannian Manifolds of Negative Curvature (in Russian), Tr. Mat. Inst. Steklova 90, Russ. Acad. Sci., Moscow, 1967; English translation in Proceedings of the Steklov Institute of Mathematics 90, Amer. Math. Soc., Providence, 1969.
  • [2] D. V. Anosov and A. B. Katok, New examples in smooth ergodic theory: Ergodic diffeomorphisms (in Russian), Tr. Mosk. Mat. Obs. 23 (1970), 3–36; English translation in Trans. Moscow Math. Soc. 23 (1970), 1–35.
  • [3] D. V. Anosov and Ya. G. Sinai, Certain smooth ergodic systems (in Russian), Uspekhi Mat. Nauk 22 (1967), 107–172; English translation in Russian Math. Surveys 22 (1967), 103–167.
  • [4] M.-C. Arnaud, Le “closing lemma” en topologie C1, Mém. Soc. Math. Fr. (N.S.) 74 (1998), 1–120.
  • [5] A. Avila, On the regularization of conservative maps, Acta Math. 205 (2010), 5–18.
  • [6] A. Avila, J. Bochi, and A. Wilkinson, Nonuniform center bunching and the genericity of ergodicity among C1 partially hyperbolic symplectomorphisms, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 931–979.
  • [7] A. T. Baraviera and C. Bonatti, Removing zero Lyapunov exponents, Ergodic Theory Dynam. Systems 23 (2003), 1655–1670.
  • [8] L. Barreira and Ya. Pesin, “Lectures on Lyapunov exponents and smooth ergodic theory,” with appendices by M. Brin and by D. Dolgopyat, H. Hu, and Ya. Pesin, in Smooth Ergodic Theory and its Applications (Seattle, 1999), Proc. Sympos. Pure Math. 69, Amer. Math. Soc., Providence, 2001, 3–106.
  • [9] L. Barreira and Ya. Pesin, Lyapunov Exponents and Smooth Ergodic Theory, Univ. Lecture Ser. 23, Amer. Math. Soc., Providence, 2002.
  • [10] J. Bochi and M. Viana, The Lyapunov exponents of generic volume-preserving and symplectic maps, Ann. of Math. (2) 161 (2005), 1423–1485.
  • [11] C. Bonatti and L. J. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math. (2) 143 (1996), 357–396.
  • [12] C. Bonatti and L. J. Díaz, Robust heterodimensional cycles and C1-generic dynamics, J. Inst. Math. Jussieu 7 (2008), 469–525.
  • [13] C. Bonatti, L. J. Díaz, and E. R. Pujals, A C1-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math. (2) 158 (2003), 355–418.
  • [14] C. Bonatti, L. J. Díaz, and M. Viana, Dynamics beyond uniform hyperbolicity: A global geometric and probabilistic perspective, Encyclopaedia Math. Sci. 102, Springer, Berlin, 2005.
  • [15] M. Brin and Ya. Pesin, Partially hyperbolic dynamical systems (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 170–212; English translation in Math. USSR-Isv. 8 (1974), 177–218.
  • [16] K. Burns, D. Dolgopyat, and Ya. Pesin, Partial hyperbolicity, Lyapunov exponents and stable ergodicity, J. Stat. Phys. 108 (2002), 927–942.
  • [17] K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math. (2) 171 (2010), 451–489.
  • [18] D. Dolgopyat and A. Wilkinson, “Stable accessibility is C1 dense” in Geometric Methods in Dynamics, II, Astérisque 287, Soc. Math. France, Paris, 2003, 33–60.
  • [19] M. Grayson, C. Pugh, and M. Shub, Stably ergodic diffeomorphisms, Ann. of Math. (2) 140 (1994), 295–329.
  • [20] E. Hopf, Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung, Ber. Verh. Sächs. Akad. Wiss. Leipzig 91 (1939), 261–304.
  • [21] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publ. Math. Inst. Hautes Études Sci. 51 (1980), 137–173.
  • [22] A. Katok and B. Haasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia Math. Appl. 54, Cambridge Univ. Press, Cambridge, 1995.
  • [23] 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.
  • [24] C. Liang, W. Sun, and J. Yang, Some results on perturbations to Lyapunov exponents, preprint, arXiv:1011.5299v1 [math.DS]
  • [25] R. Mañé, Ergodic theory and differentiable dynamics, Ergeb. Math. Grenzgeb. (3) 8, Springer, Berlin, 1987.
  • [26] Ya. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory (in Russian), Uspekhi Mat. Nauk 32 (1977), 55–112; English translation in Russian Math. Surveys 32 (1977), 55–114.
  • [27] C. Pugh and M. Shub, Ergodic attractors, Trans. Amer. Math. Soc. 312, no. 1 (1989), 1–54.
  • [28] C. Pugh and M. Shub, “Stable ergodicity and partial hyperbolicity” in International Conference on Dynamical Systems (Montevideo, 1995), Pitman Res. Notes Math. Ser. 362, Longman, Harlow, 1996, 182–187.
  • [29] C. Pugh and M. Shub, Stable ergodicity and julienne quasi-conformality, J. Eur. Math. Soc. (JEMS) 2 (2000), 1–52.
  • [30] F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi, and R. Ures, Creation of blenders in the conservative setting, Nonlinearity 23 (2010), 201–223.
  • [31] F. Rodriguez Hertz, M. A. Rodriguez Hertz, and R. Ures, “A survey of partially hyperbolic dynamics” in Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow (Toronto, 2006), Fields Inst. Commun. 51, Amer. Math. Soc., Providence, 2007, 35–87.
  • [32] F. Rodriguez Hertz, M. A. Rodriguez Hertz, and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Invent. Math. 172 (2008), 353–381.
  • [33] V. A. Rohlin, On the Fundamental Ideas of Measure Theory, Amer. Math. Soc. Transl. Ser. 2 1952, Amer. Math. Soc., Providence, 1952.
  • [34] M. Shub and A. Wilkinson, Pathological foliations and removable zero exponents, Invent. Math. 139 (2000), 495–508.
  • [35] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. (N.S.) 73 (1967), 747–817.
  • [36] A. Tahzibi, Stably ergodic diffeomorphisms which are not partially hyperbolic, Israel J. Math. 142 (2004), 315–344.