Advertisement

Communications in Mathematical Physics

, Volume 323, Issue 3, pp 975–1005 | Cite as

Embedding of Analytic Quasi-Periodic Cocycles into Analytic Quasi-Periodic Linear Systems and its Applications

  • Jiangong You
  • Qi Zhou
Article

Abstract

In this paper, we prove that any analytic quasi-periodic cocycle close to constant is the Poincaré map of an analytic quasi-periodic linear system close to constant, which bridges both methods and results in quasi-periodic linear systems and cocycles. We also show that the almost reducibility of an analytic quasi-periodic linear system is equivalent to the almost reducibility of its corresponding Poincaré cocycle. By the local embedding theorem and the equivalence, we transfer the recent local almost reducibility results of quasi-periodic linear systems (Hou and You, in Invent Math 190:209–260, 2012) to quasi-periodic cocycles, and the global reducibility results of quasi-periodic cocycles (Avila, in Almost reducibility and absolute continuity, 2010; Avila et al., in Geom Funct Anal 21:1001–1019, 2011) to quasi-periodic linear systems. Finally, we give a positive answer to a question of Avila et al. (Geom Funct Anal 21:1001–1019, 2011) and use it to study point spectrum of long-range quasi-periodic operator with Liouvillean frequency. The embedding also holds for some nonlinear systems.

Keywords

Bounded Linear Operator Implicit Function Theorem Rotation Number Anderson Localization Reducibility Result 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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), Bristol: Hilger, 1980, pp. 133–164Google Scholar
  2. 2.
    Avila A.: Density of positive Lyapunov exponents for quasiperiodic \({SL(2, \mathbb{R})}\) cocycles in arbitrary dimension. J. Mod. Dyn. 3, 629–634 (2009)Google Scholar
  3. 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/0905.3902v1 [math.DS], 2009
  4. 4.
    Avila, A.: Almost reducibility and absolute continuity. Preprint, http://arxiv.org/abs/1006.0704v1 [math.DS], 2010
  5. 5.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Avila A., Jitomirskaya S.: The Ten Martini Problem. Ann. Math. 170, 303–342 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Avila A., Jitomirskaya S.: Almost localization and almost reducibility. J. Eur. Math. Soc. 12, 93–131 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Avila A., Krikorian R.: educibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles. Ann. Math. 164, 911–940 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Berti M., Biasco L.: Forced vibrations of wave equations with non-monotone nonlinearities. Ann. I. H. Poincaré-AN. 23, 439–474 (2006)MathSciNetCrossRefzbMATHADSGoogle Scholar
  10. 10.
    Bourgain J.: On the spectrum of lattice Schrodinger operators with deterministic potential II. Dedicated to the memory of Tom Wolff. J. Anal. Math. 88, 221–254 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    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
  12. 12.
    Bourgain J., Goldstein M.: On nonperturbative localization with quasiperiodic potential. Annals of Math. 152, 835–879 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bourgain J., Jitomirskaya S.: Absolutely continuous spectrum for 1D quasiperiodic operators. Invent. Math. 148(3), 453–463 (2002)MathSciNetCrossRefzbMATHADSGoogle Scholar
  14. 14.
    Chavaudret C.: Reducibility of quasiperiodic cocycles in linear Lie groups. Erg. Th. Dyn. Sys. 31(03), 741–769 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chavaudret, C.: Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles. http://arxiv.org/abs/0912.4814v3 [math.DS], 2010 to appear in Bulletin de la Société Mathématique de France
  16. 16.
    Deimling, K.: Nonlinear functional analysis. Berlin-Heidelberg-NewYork: Springer-Verlag, 1985Google Scholar
  17. 17.
    Dias J.L.: A normal form theorem for Brjuno skew systems through renormalization. J. Diff. Eq. 230, 1–23 (2006)CrossRefzbMATHADSGoogle Scholar
  18. 18.
    Dinaburg E., Sinai Ya.: The one-dimensional Schrödinger equation with a quasi-periodic potential. Funct. Anal. Appl. 9, 279–289 (1975)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Eliasson H.: Floquet solutions for the one-dimensional quasiperiodic Schrödinger equation. Commun. Math. Phys. 146, 447–482 (1992)MathSciNetCrossRefzbMATHADSGoogle Scholar
  20. 20.
    Fayad, B., Katok, A., Windsor, A.: Mixed spectrum reparameterizations of linear flows on \({\mathbb{T}^2}\) . Dedicated to the memory of I. G. Petrovskii on the occasion of his 100th anniversary. Mosc. Math. J. 1(4), 521–537. (2001)Google Scholar
  21. 21.
    Fayad B., Krikorian R.: Rigidity results for quasiperiodic \({{\rm SL}(2,\mathbb{R})}\) -cocycles. J. Mod. Dyn. 3(4), 479–510 (2009)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Gordon A.Y.: The point spectrum of one-dimensional Schrödinger operator (Russian). Usp. Mat. Nauk. 31, 257–258 (1976)zbMATHGoogle Scholar
  23. 23.
    Gordon A.Y., Jitomirskaya S., Last Y., Simon B.: Duality and singular continuous spectrum in the almost Mathieu equation. Acta Math. 178, 169–183 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Hou X., You J.: Almost reducibility and non-perturbative reducibility of quasiperiodic linear systems. Invent. Math. 190, 209–260 (2012)MathSciNetCrossRefzbMATHADSGoogle Scholar
  25. 25.
    Jitomirskaya S.: Metal-Insulator transition for the almost Mathieu operator. Ann. Math. 150, 1159–1175 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Johnson R.: Analyticity of spectral subbundles. J. Diff. Eqs. 35(3), 366–387 (1980)CrossRefzbMATHADSGoogle Scholar
  27. 27.
    Johnson R., Sell G.: Smoothness of spectral subbundles and reducibility of quasiperiodic linear differential systems. J. Diff. Eqs. 41(2), 262–288 (1981)MathSciNetCrossRefzbMATHADSGoogle Scholar
  28. 28.
    Johnson R., Moser J.: The rotation number for almost periodic potentials. Commun. Math. Phys. 84(3), 403–438 (1982)MathSciNetCrossRefzbMATHADSGoogle Scholar
  29. 29.
    Kotani, S.: Lyaponov indices determine absolutely continuous spectra of stationary random onedimensional Schrondinger operators. In: Stochastic Analysis, K. Ito, ed., Amsterdam: North Holland, 1984, pp. 225–248Google Scholar
  30. 30.
    Krikorian R.: Global density of reducible quasi-periodic cocycles on \({\mathbb{T}^1\times SU(2)}\) . Ann. Math. 2, 269–326 (2001)MathSciNetCrossRefGoogle Scholar
  31. 31.
    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/0402333v1 [math.DS], 2004
  32. 32.
    Krikorian, R., Wang, J., You, J., Zhou, Q.: Linearization of quasiperiodically forced circle flow beyond Brjuno condition. PreprintGoogle Scholar
  33. 33.
    Kuksin, S., Pöschel, J.: On the Inclusion of Analytic Symplectic Maps in Analytic Hamiltonian Flows and Its Applications. In: Seminar on Dynamical Systems, S. Kuksin, V. Lazutkin, J. Pöchel eds., Basel: Birkhäuser, 1994, pp. 96–116Google Scholar
  34. 34.
    Maslov V.P., Molchanov S.A., Gordon A.Y.: Behavior of generalized eigenfunctions at infinity and the Schrödinger conjecture. Russ. J. Math. Phys. 1, 71–104 (1993)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Moser J., Poschel J.: An extension of a result by Dinaburg and Sinai on quasiperiodic potentials. Comment. Math. Helv. 59, 39–85 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Puig J.: A nonperturbative Eliasson’s reducibility theorem. Nonlinearity 19(2), 355–376 (2006)MathSciNetCrossRefzbMATHADSGoogle Scholar
  37. 37.
    Rychlik M.: Renormalization of cocycles and linear ODE with almost-periodic coefficients. Invent. Math. 110, 173–206 (1992)MathSciNetCrossRefzbMATHADSGoogle Scholar
  38. 38.
    Simon B.: Kotani theory for one-dimensional stochastic Jacobi matrices. Commun. Math. Phys. 89, 227–234 (1983)CrossRefzbMATHADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsNanjing UniversityNanjingChina
  2. 2.Laboratoire de Probabilités et Modèles aléatoiresUniversité Pierre et Marie CurieParis Cedex 05France

Personalised recommendations