• A. AVILA
• R. KRIKORIAN

Code(s) de Classification MSC:

• 47B39 Difference operators [See also 39A70]
• 47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
• 37H15 Multiplicative ergodic theory, Lyapunov exponents [See also 34D08, 37Axx, 37Cxx, 37Dxx]
• 35B15 Almost periodic solutions

Résumé: We show that for almost every frequency $\alpha \in \R \setminus \Q$, for every $C^\omega$ potential $v:\R/\Z \to \R$, and for almost every energy $E$ the corresponding quasiperiodic Schr\"odinger cocycle is either reducible or non-uniformly hyperbolic. This result gives a very good control on the absolutely continuous part of the spectrum of the corresponding quasiperiodic Schr\"odinger operator, and allows us to complete the proof of the Aubry-Andr\'e conjecture on the measure of the spectrum of the Almost Mathieu Operator.

Mots Clés: Quasi-periodic Schrödinger operator ; spectrum ; Lyapunov exponents ; Floquet reducibility ; renormalization

Date: 2004-03-10

Prépublication numéro: PMA-891