Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles

Artur Avila; Raphaël Krikorian

doi:10.4007/annals.2006.164.911

Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles

Artur Avila
, Raphaël Krikorian

Collège de France
Sorbonne Université

Research output: Contribution to journal › Article › peer-review

150 Citations (Scopus)

Abstract

We show that for almost every frequency α ∈ ℝ\ℚ, for every C^ω potential v : ℝ/ℤ → ℝ, and for almost every energy E the corresponding quasiperiodic Schrödinger cocycle is either reducible or nonuniformly hyperbolic. This result gives very good control on the absolutely continuous part of the spectrum of the corresponding quasiperiodic Schrödinger operator, and allows us to complete the proof of the Aubry-André conjecture on the measure of the spectrum of the Almost Mathieu Operator.

Original language	English
Pages (from-to)	911-940
Number of pages	30
Journal	Annals of Mathematics
Volume	164
Issue number	3
DOIs	https://doi.org/10.4007/annals.2006.164.911
Publication status	Published - 1 Jan 2006

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10.4007/annals.2006.164.911

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title = "Reducibility or nonuniform hyperbolicity for quasiperiodic Schr{\"o}dinger cocycles",

abstract = "We show that for almost every frequency α ∈ ℝ\textbackslash{}ℚ, for every Cω potential v : ℝ/ℤ → ℝ, and for almost every energy E the corresponding quasiperiodic Schr{\"o}dinger cocycle is either reducible or nonuniformly hyperbolic. This result gives very good control on the absolutely continuous part of the spectrum of the corresponding quasiperiodic Schr{\"o}dinger 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.",

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AU - Krikorian, Raphaël

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N2 - We show that for almost every frequency α ∈ ℝ\ℚ, for every Cω potential v : ℝ/ℤ → ℝ, and for almost every energy E the corresponding quasiperiodic Schrödinger cocycle is either reducible or nonuniformly hyperbolic. This result gives very good control on the absolutely continuous part of the spectrum of the corresponding quasiperiodic Schrödinger operator, and allows us to complete the proof of the Aubry-André conjecture on the measure of the spectrum of the Almost Mathieu Operator.

AB - We show that for almost every frequency α ∈ ℝ\ℚ, for every Cω potential v : ℝ/ℤ → ℝ, and for almost every energy E the corresponding quasiperiodic Schrödinger cocycle is either reducible or nonuniformly hyperbolic. This result gives very good control on the absolutely continuous part of the spectrum of the corresponding quasiperiodic Schrödinger operator, and allows us to complete the proof of the Aubry-André conjecture on the measure of the spectrum of the Almost Mathieu Operator.

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DO - 10.4007/annals.2006.164.911

M3 - Article

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SN - 0003-486X

VL - 164

SP - 911

EP - 940

JO - Annals of Mathematics

JF - Annals of Mathematics

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ER -

Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles

Abstract

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