Communications in Mathematical Physics

, Volume 332, Issue 1, pp 189–219 | Cite as

Positive Lyapunov Exponents for Higher Dimensional Quasiperiodic Cocycles

  • Pedro Duarte
  • Silvius KleinEmail author


We consider an m-dimensional analytic cocycle \({\mathbb{T} \times \mathbb{R}^m \ni (x, \vec{\psi}) \mapsto (x + \omega, A (x) \cdot \vec{\psi}) \in \mathbb{T} \times \mathbb{R}^m}\), where \({\omega \notin \mathbb{Q}}\) and \({A \in C^\omega (\mathbb{T}, \mathrm{Mat}_m (\mathbb{R}))}\). Assuming that the d × d upper left corner block of A is typically large enough, we prove that the d largest Lyapunov exponents associated with this cocycle are bounded away from zero. The result is uniform relative to certain measurements on the matrix blocks forming the cocycle. As an application of this result, we obtain nonperturbative (in the spirit of Sorets–Spencer theorem) positive lower bounds of the nonnegative Lyapunov exponents for various models of band lattice Schrödinger operators.


Lyapunov Exponent Transversality Condition Subharmonic Function Large Lyapunov Exponent Block Matrice 
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  1. 1.
    Bourgain, J.: Green’s function estimates for lattice Schrödinger operators and applications. In: Annals of Mathematics Studies, vol. 158, Princeton University Press, Princeton (2005)Google Scholar
  2. 2.
    Bourgain, J., Jitomirskaya, S.: Anderson localization for the band model, Geometric aspects of functional analysis. In: Lecture Notes in Mathematics, vol. 1745, pp. 67–79. Springer, Berlin (2000)Google Scholar
  3. 3.
    Bruns, W., Vetter, U.: Determinantal rings. In: Lecture Notes in Mathematics, vol. 1327. Springer, Berlin (1988)Google Scholar
  4. 4.
    Chan J.: Method of variations of potential of quasi-periodic Schrödinger equations. Geom. Funct. Anal. 17(5), 1416–1478 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Duren, Peter L.: Theory of H p spaces. In: Pure and Applied Mathematics, vol. 38. Academic Press, New York (1970)Google Scholar
  6. 6.
    Federer, H.; Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer, New York (1969)Google Scholar
  7. 7.
    Gol’dsheĭd I.Ya., Sorets E.: Lyapunov exponents of the Schrödinger equation with quasi-periodic potential on a strip. Commun. Math. Phys. 145(3), 507–513 (1992)ADSCrossRefGoogle Scholar
  8. 8.
    Herman M.-R.: Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnol’d et de Moser sur le tore de dimension 2. Comment. Math. Helv. 58(3), 453–502 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Hunt B.R., Sauer T., Yorke A.J.: Prevalence: a translation-invariant “almost every” on infinite-dimensional spaces. Bull. Am. Math. Soc. (N.S.) 27(2), 217–238 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Kotani S., Simon B.: Stochastic Schrödinger operators and Jacobi matrices on the strip. Commun. Math. Phys. 119(3), 403–429 (1988)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Levin, B.Ya.: Lectures on entire functions, Translations of Mathematical Monographs, vol. 150, American Mathematical Society, Providence, RI, (1996), in collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko, translated from the Russian manuscript by Tkachenko.Google Scholar
  12. 12.
    Schlag W.: Regularity and convergence rates for the Lyapunov exponents of linear co-cycles. J. Mod. Dyn. 7(4), 619–637 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Sorets E., Spencer T.: Positive Lyapunov exponents for Schrödinger operators with quasi-periodic potentials. Commun. Math. Phys. 142(3), 543–566 (1991)ADSCrossRefzbMATHMathSciNetGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Departamento de Matemática and CMAF, Faculdade de CiênciasUniversidade de LisboaLisboaPortugal
  2. 2.CMAF, Faculdade de CiênciasUniversidade de LisboaLisboaPortugal
  3. 3.IMARBucharestRomania

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