Skip to main content
SpringerLink
Log in

Reducibility Results for Quasiperiodic Cocycles With Liouvillean Frequency

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

We study the reducibility problems for quasiperiodic cocycles in linear Lie groups with one frequency, irrespective of any Diophantine condition on the base dynamics. Under a non-degeneracy condition, a positive measure diagonalizable result is obtained for quasiperiodic \({GL(d,\mathbb R)}\) cocycles which are close to constants. It generalizes previous works by Avila–Fayad–Krikorian and Hou–You, and our approach is based on periodic approximation and KAM schemes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aubry, S., André, G.: Analyticity breaking and Anderson localization in incommensurate lattices. In: Group Theoretical Methods in Physics (Proc. Eighth Internat. Colloq. Kiryat Anavim, 1979), Hilger, Bristol, pp. 133–164 (1980)

  2. Avila, A.: Almost reducibility and absolute continuity, preprint. http://arxiv.org/abs/1006.0704 (2010)

  3. Avila, A.: Global theory of one-frequency Schrödinger operators I: stratified analyticity of the Lyapunov exponent and the boundary of nonuniform hyperbolicity, preprint. http://arxiv.org/abs/0905.3902 (2009)

  4. Avila, A.: Global theory of one-frequency Schrödinger operators II: acriticality and finiteness of phase transitions for typical potentials, preprint (2010)

  5. Avila A., Jitomirskaya S.: Almost localization and almost reducibility. J. Eur. Math. Soc. 12, 93–131 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Avila A., Jitomirskaya S.: The ten Martini problem. Ann. Math. 170, 303–342 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Avila A., Krikorian R.: Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles. Ann. Math. 164, 911–940 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Avila A., Fayad B., Krikorian R.: A KAM scheme for \({{\rm SL}(2,\mathbb R)}\) cocycles with Liouvillean frequencies. Geom. Funct. Anal. 21, 1001–1019 (2011)

    Article  MATH  Google Scholar 

  9. Bourgain J., Jitomirskaya S.: Absolutely continuous spectrum for 1D quasiperiodic operators. Invent. Math. 148(3), 453–463 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chavaudret, C.: Reducibility of quasiperiodic cocycles in linear Lie groups. Ergod. Theory Dyn. Syst. (2010). doi:10.1017/S0143385710000076

  11. Damanik D.: Lyapunov exponents and spectral analysis of ergodic Schrödinger operators: a survey of Kotani theory and its applications. Spectral Theory Mathematical Physics: a Festschrift in honor of Barry Simons 60th Birthday. Proc. Symp. Pure Math. 76(2), 539–563 (2010)

    MathSciNet  Google Scholar 

  12. Dinaburg E., Sinai Ya.: The one-dimensional Schrödinger equation with a quasi-periodic potential. Funct. Anal. Appl. 9, 279–289 (1975)

    Article  MathSciNet  Google Scholar 

  13. Eliasson H.: Floquet solutions for the one-dimensional quasiperiodic Schrödinger equation. Commun. Math. Phys. 146, 447–482 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fayad B., Krikorian R.: Rigitidy results for quasiperiodic \({{\rm SL}(2, \mathbb R)}\) -cocycles. J. Mod. Dyn. 3(4), 479–510 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Her H., You J.: Full measure reducibility for generic one-parameter family of quasiperiodic linear systems. J. Dyn. Differ. Equ. 20(4), 831–866 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hou, X., You, J.: Almost reducibility and non-perturbative reducibility of quasiperiodic linear systems, preprint (2010)

  17. Jorba A., Simó C.: On the reducibility of linear differential equations with quasi-periodic coeffcients. J. Differ. Equ. 98, 111–124 (1992)

    Article  MATH  Google Scholar 

  18. Kotani S.: Lyaponov indices determine absolutely continuous spectra of stationary random onedimensional Schrondinger operators. In: Ito, K. (eds) Stochastic Analysis, pp. 225–248. North Holland, Amsterdam (1984)

    Google Scholar 

  19. Krikorian R.: Réductibilité presque partout des flots fibrés quasi-périodiques à valeurs dans les groupes compacts. Ann. Scient. Éc. Norm. Sup. 32, 187–240 (1999a)

    MathSciNet  MATH  Google Scholar 

  20. Krikorian R.: Réductibilité des systèmes produits-croisés à valeurs das des groupes compacts. Astérisque 259, 1–216 (1999b)

    Google Scholar 

  21. Krikorian R.: Global density of reducible quasi-periodic cocycles on \({\mathbb{T}^1\times SU(2)}\) . Ann. Math. 2, 269–326 (2001)

    Article  MathSciNet  Google Scholar 

  22. Krikorian, R.: Reducibility, differentiable rigidity and Lyapunov exponents for quasiperiodic cocycles on \({\mathbb{T}\times SL(2,\mathbb{R})}\) , preprint. http://arxiv.org/abs/math/0402333 (2004)

  23. Krikorian, R., Wang, J., You, J., Zhou, Q.: Linearization of quasiperiodically forced circle flow beyond Brjuno condition, in preparation (2011)

  24. Moser J., Poschel J.: An extension of a result by Dinaburg and Sinai on quasiperiodic potentials. Comment. Math. Helv 59, 39–85 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  25. Puig J.: A nonperturbative Eliasson’s reducibility theorem. Nonlinearity 19(2), 355–376 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Rüssmann H.: On the one dimensional Schrödinger equation with a quasi-periodic potential. Ann. N.Y. Acad. Sci. 357, 90–107 (1980)

    Article  Google Scholar 

  27. Rychlik M.: Renormalization of cocycles and linear ODE with almost-periodic coefficients. Invent. Math. 110, 173–206 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  28. Simon B.: Kotani theory for one-dimensional stochastic Jacobi matrices. Commun. Math. Phys. 89, 227–234 (1983)

    Article  MATH  Google Scholar 

  29. Xu J., You J., Qiu Q.: Invariant tori for nearly integrable Hamiltonian systems with degeneracy. Math. Z. 226, 375–387 (1997)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, 210093, China

    Qi Zhou & Jing Wang

Authors
  1. Qi Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jing Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qi Zhou.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhou, Q., Wang, J. Reducibility Results for Quasiperiodic Cocycles With Liouvillean Frequency. J Dyn Diff Equat 24, 61–83 (2012). https://doi.org/10.1007/s10884-011-9235-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-011-9235-0

Keywords

Search

Navigation