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Communications in Mathematical Physics

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

Positive Lyapunov Exponents for Higher Dimensional Quasiperiodic Cocycles

Article

Abstract

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.

Keywords

Lyapunov Exponent Transversality Condition Subharmonic Function Large Lyapunov Exponent Block Matrice 
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.

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Copyright information

© 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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