Communications in Mathematical Physics

, Volume 362, Issue 3, pp 801–826 | Cite as

The Set of Smooth Quasi-periodic Schrödinger Cocycles with Positive Lyapunov Exponent is Not Open

  • Yiqian Wang
  • Jiangong YouEmail author


One knows that the set of quasi-periodic Schrödinger cocycles with positive Lyapunov exponent is open and dense in analytic topology. In this paper, we construct cocycles with positive Lyapunov exponent which can be arbitrarily approximated by ones with zero Lyapunov exponent in the space of \({\mathcal{C}^ l (1 \le l \le \infty)}\) smooth quasi-periodic cocycles, which shows that the set of quasi-periodic Schrödinger cocycles with positive Lyapunov exponent is not open in smooth topology.


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We are grateful to the referee for the useful suggestions. We are also in debt to S. Jitomirskaya for drawing our attention to this question.


  1. 1.
    Avila A.: Density of positive Lyapunov exponents for \({SL(2,\mathbb{R})}\) cocycles. J. Am. Math. Soc. 24, 999–1014 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Avila A.: Global theory of one-frequency Schrödinger operators. Acta Math. 215, 1–54 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Avila A., Jitomirskaya S.: The ten Martini problem. Ann. Math. 170, 303–342 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Avila A., Jitomirskaya S., Sadel C.: Complex one-frequency cocycles. J. Eur. Math. Soc. 16(9), 1915–1935 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Avila A., Krikorian R.: Monotonic cocycles. Invent. Math. 202, 271–331 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Avila A., Viana M.: Extremal Lyapunov exponents: an invariance principle and applications. Inventiones Math. 181, 115–189 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Benedicks M., Carleson L.: The dynamics of the Hénon map. Ann. Math. 133, 73–169 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bjerklöv K.: Explicit examples of arbitrarily large analytic ergodic potentials with zero Lyapunov exponent. Geom. Funct. Anal. 16(6), 1183–1200 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bjerklöv, K.: The dynamics of a class of quasi-periodic Schrödinger cocycles. Ann. Henri Poincaré 16(4), 961–1031 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bocher-Neto, C., Viana, M.: Continuity of Lyapunov exponents for random 2D matrices. arXiv:1012.0872v1 (2010)
  11. 11.
    Bochi J.: Genericity of zero Lyapunov exponents. Ergod. Theory Dyn. Syst. 22(6), 1667–1696 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bochi J., Fayad B.: Dichotomies between uniform hyperbolicity and zero Lyapunov exponents for \({SL(2,\mathbb{R})}\) cocycles. Bull. Braz. Math. Soc. New Ser. 37(3), 307–349 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bochi J., Viana M.: The Lyapunov exponents of generic volume perserving and symplectic maps. Ann. Math. 161, 1–63 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Bonatti C., Gómez-Mont X., Viana M.: Généricité d’exposants de Lyapunov non-nuls pour des produits déterministes de matrices. Ann. Inst. Henri Poincaré Anal. Non Linéaire 20, 579–624 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bourgain J.: Green’s Function Estimates for Lattice Schrödinger Operators and Applications. Annals of Mathematics Studies, 158. Princeton University Press, Princeton (2005)CrossRefGoogle Scholar
  16. 16.
    Bourgain J.: Positivity and continuity of the Lyapunov exponent for shifts on \({\mathbb{T}^d}\) with arbitrary frequency vector and real analytic potential. J. Anal. Math. 96, 313–355 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Bourgain J., Goldstein M.: On nonperturbative localization with quasi-periodic potential. Ann. Math. 152, 835–879 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Bourgain J., Goldstein M., Schlag W.: Anderson localization for Schrödinger operators on \({\mathbb{Z}}\) with potentials given by skew-shift. Commun. Math. Phys. 220(3), 583–621 (2001)ADSCrossRefzbMATHGoogle Scholar
  19. 19.
    Bourgain J., Jitomirskaya S.: Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential. J. Stat. Phys. 108(5–6), 1203–1218 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Bourgain J., Schlag W.: Anderson localization for Schrödinger operators on \({\mathbb{Z}}\) with strongly mixing potentials. Commun. Math. Phys. 215, 143–175 (2000)ADSCrossRefzbMATHGoogle Scholar
  21. 21.
    Duarte P., Klein S.: Continuity of the Lyapunov exponents for quasiperiodic cocycles. Commun. Math. Phys. 332(3), 1113–1166 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Eliasson L.H.: Discrete one-dimensional quasi-periodic Schrödinger operators with pure point spectrum. Acta Math. 179(2), 153–196 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Fröhlich J., Spencer T., Wittwer P.: Localization for a class of one-dimensional quasi-periodic Schrödinger operators. Commun. Math. Phys. 132, 5–25 (1990)ADSCrossRefzbMATHGoogle Scholar
  24. 24.
    Furman A.: On the multiplicative ergodic theorem for the uniquely ergodic systems. Ann. Inst. Henri Poincaré 33, 797–815 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Furstenberg H.: Noncommuting random products. Trans. Am. Math. Soc. 108, 377–428 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Furstenberg H., Kifer Y.: Random matrix products and measures in projective spaces. Isr. J. Math 10, 12–32 (1983)CrossRefzbMATHGoogle Scholar
  27. 27.
    Goldstein M., Schlag W.: Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions. Ann. Math. 154, 155–203 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Hennion H.: Loi des grands nombres et perturbations pour des produits réductibles de matrices aléatoires indépendantes. Z. Wahrsch. Verw. Gebiete 67, 265–278 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Herman M.: Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnold et de Moser sur le tore de dimension 2. Comment. Math. Helv. 58, 453–502 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Ishii K.: Localization of eigenstates and transport phenomena in one-dimensional disordered systems. Suppl. Prog. Theor. Phys. 53, 77–138 (1973)ADSCrossRefGoogle Scholar
  31. 31.
    Jitomirskaya S., Koslover D., Schulteis M.: Continuity of the Lyapunov exponent for general analytic quasiperiodic cocycles. Ergod. Theory Dyn. Syst. 29, 1881–1905 (2009)CrossRefzbMATHGoogle Scholar
  32. 32.
    Jitomirskaya S., Marx C.: Continuity of the Lyapunov exponent for analytic quasi-periodic cocycles with singularities. Journal of Fixed Point Theory and Applications 10, 129–146 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Jitomirskaya S., Marx C.: Analytic quasi-perodic cocycles with singularities and the Lyapunov exponent of extended Harper’s model. Comm. Math. Phys. 316(1), 237–267 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Jitomirskaya S., Marx C.: Analytic quasi-periodic Schrödinger operators and rational frequency approximants. Geom. Funct. Anal. 22(5), 1407–1443 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Jitomirskaya S., Mavi R.: Continuity of the measure of the spectrum for quasiperiodic Schrödinger operators with rough potentials. Commun. Math. Phys. 325(2), 585–601 (2014)ADSCrossRefzbMATHGoogle Scholar
  36. 36.
    Jitomirskaya, S., Mavi R.: Dynamical bounds for quasiperiodic Schrödinger operators with rough potentials. arXiv:1412.0309 (2014)
  37. 37.
    Kotani, S.: Ljapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators, Stochastic Analysis(Katata/Kyoto, 1982), (North-Holland Math. Library 32, North-Holland, Amsterdam), 225–247 (1984)Google Scholar
  38. 38.
    Klein S.: Anderson localization for the discrete one-dimensional quasi-periodic Schrödinger operator with potential defined by a \({{{\mathcal{C} }}^{\infty}}\)-class function. Journal of Functional Analysis 218(2), 255–292 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Knill, O.: The upper Lyapunov exponent of \({SL(2,\mathbb{R})}\) cocycles: Discontinuity and the problem of positivity, Lecture notes in Math. 1486, Lyapunov exponents (Oberwolfach, 1990) 86–97, (1991)Google Scholar
  40. 40.
    Kotani S.: Jacobi matrices with random potentials taking finitely many values. Rev. Math. Phys. 1, 129–133 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Mañé, R.: Oseledec’s theorem from the generic viewpoint, In: Proceedings of the ICM (Warsaw, 1983), 1269–1276, PWN, Warsaw, (1984)Google Scholar
  42. 42.
    Mañé, R.: The Lyapunov exponents of generic area preserving diffeomorphisms, In International Conference on Dynamical Systems (Montevideo, 1995), 110–119, Pitman Res. Notes Math. 362, Longman, Harlow, (1996)Google Scholar
  43. 43.
    Malheiro E.C., Viana M.: Lyapunov exponents of linear cocycles over Markov shifts. Stoch. Dyn. 15(3), 1550020 (2015)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Pastur L.A.: Spectral properties of disordered systems in one-body approximation. Comm. Math. Phys. 75, 179–196 (1980)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Sinai Ya.G.: Anderson localization for one-dimensional difference Schrödinger operator with quasiperiodic potential. J. Statist. Phys. 46, 861–909 (1987)ADSMathSciNetCrossRefGoogle Scholar
  46. 46.
    Sorets E., Spencer T.: Positive Lyapunov exponents for Schrödinger operators with quasi-periodic potentials. Commun. Math. Phys. 142, 543–566 (1991)ADSCrossRefzbMATHGoogle Scholar
  47. 47.
    Thouvenot J.: An example of discontinuity in the computation of the Lyapunov exponents. Proc. Stekolov Inst. Math. 216, 366–369 (1997)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Viana M.: Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents. Annals of Mathematics 167, 643–680 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Viana M., Yang J.: Physical measures and absolute continuity for one-dimensional center direction. Annales Inst. H. Poincaré-Analyse Non-Linéaire 30, 845–877 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Wang Y., You J.: Examples of discontinuity of Lyapunov exponent in smooth quasi-periodic cocycles. Duke Math. J. 162, 2363–2412 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Wang Y., Zhang Z.: Uniform positivity and continuity of Lyapunov exponents for a class of \({C^2}\) quasiperiodic Schrödinger cocycles. J. Funct. Anal. 268, 2525–2585 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    You J., Zhang S.: Hölder continuity of the Lyapunov exponent for analytic quasiperiodic Schröinger cocycle with weak Liouville frequency. Ergod. Theory Dyn. Syst. 34, 1395–1408 (2014)CrossRefzbMATHGoogle Scholar
  53. 53.
    Young L.: Lyapunov exponents for some quasi-periodic cocycles. Ergod. Theory Dyn. Syst. 17, 483–504 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Zhang Z.: Positive Lyapunov exponents for quasiperiodic Szegö cocycles. Nonlinearity 25, 1771–1797 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsNanjing UniversityNanjingChina
  2. 2.Chern Institute of Mathematics and LPMCNankai UniversityTianjinChina

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