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Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles. (English) Zbl 1138.47033

The properties of quasiperiodic Schrödinger cocycles are studied and the conditions of reducibility and nonuniform hyperbolicity are established. A one-dimensional quasiperiodic cocycle is considered as a pair $\left(\alpha ,A\right)\in ℝ×{C}^{r}\left(ℝ/ℤ,SL\left(2,ℝ\right)\right)$ viewed as a linear skew-product: $\left(\alpha ,A\right):ℝ/ℤ×{ℝ}^{2}\to ℝ/ℤ×{ℝ}^{2}$. The Schrödinger cocycle ${S}_{v,E}\in {C}^{r}\left(ℝ/ℤ,SL\left(2,ℝ\right)\right)$ is then defined, where $v$ is called the potential and $E$ the energy. The corresponding quasiperiodic Schrödinger cocycle ${H}_{v,\alpha ,x}$ is defined and it is established that its properties are closely connected to those of the family of cocycles $\left(\alpha ,{S}_{v,E}\right)$, $E\in ℝ$. It is shown that for almost every frequency $\alpha \in ℝ/ℚ$, for every ${C}^{\omega }$ potential $v:ℝ/ℤ\to ℝ$, and for almost every energy $E$, the corresponding quasiperiodic Schrödinger cocycle is either reducible or nonuniformly hyperbolic.

Several interesting results are emphasized: (i) zero Lebesgue measure of the singular continuous part of the spectrum of $H$ for almost every frequency, (ii) persistence of absolutely continuous spectrum under perturbations of the potential, (iii) continuity of the Lebesgue measure of the spectrum of $H$, under perturbations of the potential. It is concluded that the results of the paper give very good control on the absolutely continuous part of the spectrum of the quasiperiodic Schrödinger operator. It allows also to complete the proof of the Aubry–Andre conjecture on the measure of the spectrum of the almost Mathieu operator.

##### MSC:
 47B80 Random operators (linear) 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 37A20 Orbit equivalence, cocycles, ergodic equivalence relations 37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 39A10 Additive difference equations 47B36 Jacobi (tridiagonal) operators (matrices) and generalizations 47N50 Applications of operator theory in quantum physics