Advertisement

Continuity of Lyapunov exponents in the C0 topology

  • Marcelo Viana
  • Jiagang Yang
Article

Abstract

We prove that the Bochi–Mañé theorem is false, in general, for linear cocycles over non-invertible maps: there are C0-open subsets of linear cocycles that are not uniformly hyperbolic and yet have Lyapunov exponents bounded from zero.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. Avila, J. Santamaria and M. Viana, Holonomy invariance: rough regularity and applications to Lyapunov exponents, Astérisque 358 (2013), 13–74, 2013.MathSciNetzbMATHGoogle Scholar
  2. [2]
    A. Avila and M. Viana, Extremal Lyapunov exponents: an invariance principle and applications, Inventiones Mathematicae 181 (2010), 115–189.MathSciNetCrossRefGoogle Scholar
  3. [3]
    L. Backes, A. Brown and C. Butler, Continuity of Lyapunov exponents for cocycles with invariant holonomies, Journal of Modern Dynamics 12 (2018), 223–260.CrossRefGoogle Scholar
  4. [4]
    J. Bochi, Genericity of zero Lyapunov exponents. PhD thesis, IMPA, 2000, preprint, www.preprint.impa.br.Google Scholar
  5. [5]
    J. Bochi, Genericity of zero Lyapunov exponents, Ergodic Theory and Dynamical Systems 22 (2002), 1667–1696.MathSciNetCrossRefGoogle Scholar
  6. [6]
    J. Bochi, C1-generic symplectic diffeomorphisms: partial hyperbolicity and zero centre Lyapunov exponents, Journal of the Institute of Mathematics of Jussieu 9 (2010), 49–93.MathSciNetCrossRefGoogle Scholar
  7. [7]
    J. Bochi and M. Viana, The Lyapunov exponents of generic volume-preserving and symplectic maps, Annals of Mathematics 161 (2005), 1423–1485.MathSciNetCrossRefGoogle Scholar
  8. [8]
    C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity, Encyclopaedia of Mathematical Sciences, Vol. 102, Springer-Verlag, Berlin, 2005.Google Scholar
  9. [9]
    C. Bonatti, X. Gómez-Mont and M. Viana, Généricité d’exposants de Lyapunov nonnuls pour des produits déterministes de matrices, Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 20 (2003), 579–624.Google Scholar
  10. [10]
    C. Bonatti and M. Viana, Lyapunov exponents with multiplicity 1 for deterministic products of matrices, Ergodic Theory and Dynamical Systems 24 (2004), 1295–1330.MathSciNetCrossRefGoogle Scholar
  11. [11]
    R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer Verlag, Berlin–New York, 1975.Google Scholar
  12. [12]
    C. Butler, Discontinuity of Lyapunov exponents near fiber bunched cocycles, Ergodic Theory and Dynamical Systems 38 (2018), 523–539.MathSciNetCrossRefGoogle Scholar
  13. [13]
    P. Duarte and S. Klein, Lyapunov Exponents of Linear Cocycles, Atlantis Studies in Dynamical Systems, Vol. 3, Atlantis Press, Paris, 2016.Google Scholar
  14. [14]
    H. Furstenberg, Non-commuting random products, Transactions of the American Mathematical Society 108 (1963), 377–428.MathSciNetCrossRefGoogle Scholar
  15. [15]
    H. Furstenberg and H. Kesten, Products of random matrices, Annals of Mathematical Statistics 31 (1960), 457–469.MathSciNetCrossRefGoogle Scholar
  16. [16]
    M. Herman, Une méthode nouvelle pour minorer les exposants de Lyapunov et quelques exemples montrant le caractère local d’un théorème d’Arnold et de Moser sur le tore de dimension 2, Commentarii Mathematici Helvetici 58 (1983), 453–502.MathSciNetCrossRefGoogle Scholar
  17. [17]
    B. Kalinin, Livšic theorem for matrix cocycles, Annals of Mathematics 173 (2011), 1025–1042.MathSciNetCrossRefGoogle Scholar
  18. [18]
    J. Kingman, The ergodic theory of subadditive stochastic processes, Journal of the Royal Statistical Society 30 (1968), 499–510.MathSciNetzbMATHGoogle Scholar
  19. [19]
    O. Knill, The upper Lyapunov exponent of SL(2,R) cocycles: discontinuity and the problem of positivity, in Lyapunov Exponents (Oberwolfach, 1990), Lecture Notes in Mathematics, Vol. 1486, Springer-Verlag, Berlin, 1991, pp. 86–97.Google Scholar
  20. [20]
    F. Ledrappier, Propriétés ergodiques des mesures de Sinäı, Institut des Hautes Études Scientifiques. Publications Mathématiques 59 (1984), 163–188.CrossRefGoogle Scholar
  21. [21]
    F. Ledrappier, Positivity of the exponent for stationary sequences of matrices, in Lyapunov Exponents (Bremen, 1984), Lecture Notes in Mathematics, Vol. 1186, Springer-Verlag, Berlin, 1986, pp. 56–73.MathSciNetzbMATHGoogle Scholar
  22. [22]
    F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula, Annals of Mathematics 122 (1985), 509–539.MathSciNetCrossRefGoogle Scholar
  23. [23]
    R. Ma˜né, Oseledec’s theorem from the generic viewpoint, in Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Warsaw, 1983), PWN, Warsaw, 1984, pp. 1269–1276.Google Scholar
  24. [24]
    V. I. Oseledets, A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems, Transactions of the Moscow Mathematical Society 19 (1968), 197–231.zbMATHGoogle Scholar
  25. [25]
    V. A. Rokhlin, Lectures on the entropy theory of measure-preserving transformations, Russian Mathematical Surveys 22(1967), 1–52; translated from Uspekhi Matematičeskih Nauk 22 (1967), 3–56.zbMATHGoogle Scholar
  26. [26]
    A. Tahzibi and J. Yang, Invariance principle and rigidity of high entropy measures, Transactions of the American Mathematical Society, to appear.Google Scholar
  27. [27]
    M. Viana, Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents, Annals of Mathematics 167 (2008), 643–680.MathSciNetCrossRefGoogle Scholar
  28. [28]
    M. Viana, Lectures on Lyapunov Exponents, Cambridge Studies in Advanced Mathematics, Vol. 145, Cambridge University Press, Cambridge, 2014.Google Scholar
  29. [29]
    M. Viana and K. Oliveira, Foundations of Ergodic Theory, Cambridge Studies in Advanced Mathematics, Vol. 151, Cambridge University Press, Cambridge, 2015.Google Scholar
  30. [30]
    J. Yang, Entropy along expanding foliations, arXiv:1601.05504.Google Scholar
  31. [31]
    L.-S. Young, Some open sets of nonuniformly hyperbolic cocycles, Ergodic Theory and Dynamical Systems 13 (1993), 409–415.MathSciNetCrossRefGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  1. 1.IMPARio de JaneiroBrazil
  2. 2.Departamento de Geometria, Instituto de Matemática e EstatísticaUniversidade Federal FluminenseNiteróiBrazil

Personalised recommendations