tailieunhanh - Đề tài " Reducibility or nonuniform hyperbolicity for quasiperiodic Schr¨odinger cocycles "

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 Schr¨dinger cocycle is either reducible or nonuniformly hyperbolic. This result o gives very good control on the absolutely continuous part of the spectrum of the corresponding quasiperiodic Schr¨dinger operator, and allows us to complete o the proof of the Aubry-Andr´ conjecture on the measure of the spectrum of e the Almost Mathieu Operator. . | Annals of Mathematics Reducibility or nonuniform hyperbolicity for quasiperiodic Schr odinger cocycles By Artur Avila and Rapha el Krikorian Annals of Mathematics 164 2006 911 940 Reducibility or nonuniform hyperbolicity for quasiperiodic Schrodinger cocycles By Artur Avila and Raphael Krikorian Abstract We show that for almost every frequency a G R Q for every C potential v R Z R and for almost every energy E the corresponding quasiperiodic Schrodinger 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 Schrodinger operator and allows us to complete the proof of the Aubry-Andre conjecture on the measure of the spectrum of the Almost Mathieu Operator. 1. Introduction A one-dimensional quasiperiodic Cr-cocycle in SL 2 R briefly a Cr-cocycle is a pair a A G RxCr R Z SL 2 R viewed as a linear skew-product a A R Z x R2 R Z x R2 x w x a A x w . For n G Z we let An G Cr R Z SL 2 R be defined by the rule a A n na An we will keep the dependence of An on a implicit . Thus A0 x id 0 An x JJ A x ja A x n 1 a A x for n 1 j n-1 and A-n x An x na -1. The Lyapunov exponent of a A is defined as L a A lim I ln An x dx 0. n - n JR Z Also a A is uniformly hyperbolic if there exists a continuous splitting Es x Eu x R2 and C 0 0 A 1 such that for every n 1 we have An x w CAn w w G Es x A-n x w CAn w w G Eu x . A. A. is a Clay Research Fellow. 912 ARTUR AVILA AND RAPHAEL KRIKORIAN Such splitting is automatically unique and thus invariant that is A x Es x Es x a and A x Eu x Eu x a . The set of uniformly hyperbolic cocycles is open in the C0-topology one allows perturbations both in a and in A . Uniformly hyperbolic cocycles have a positive Lyapunov exponent. If a A has positive Lyapunov exponent but is not uniformly hyperbolic then it will be called nonuniformly hyperbolic. We say that a Cr-cocycle a A is Cr-reducible if there exists B E Cr R 2Z SL

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