Almost reducibility and absolute continuity I

2010/06/03 by Artur Avila, Avila, Artur · 9 citations
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical Analysis and Transform Methods #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1006.0704

openalex publication_date 2010/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider one-frequency analytic SL(2,R) cocycles. Our main result establishes the Almost Reducibility Conjecture in the case of exponentially Liouville frequencies. Together with our earlier work, this implies that all cocycles close to constant are almost reducible, independent of the frequency. In our forthcoming work, we discuss applications to the analysis of the absolutely continuous spectrum of one-frequency Schrodinger operators.

Citations

• The absolutely continuous spectrum of the almost Mathieu operator
• Holder Continuity of the Integrated Density of States for Quasi-Periodic Schrodinger Equations and Averages of Shifts of Subharmonic Functions

Cited by

• Analytic quasi-periodic Schrödinger operators and rational frequency approximants
• Quantitative almost reducibility and Möbius disjointness for analytic quasiperiodic Schrodinger cocycles
• Mixed spectral types for the one frequency discrete quasi-periodic Schrödinger operator
• Absolutely Continuous Spectrum of Multifrequency Quasiperiodic Schrödinger operator
• Arithmetic version of Anderson localization via reducibility
• Spectral theory of the multi-frequency quasi-periodic operator with a Gevrey type perturbation
• Dry Ten Martini problem for non self-dual extended Harper's model
• Exponential decay of the size of spectral gaps for quasiperiodic\n Schr "odinger operators
• Schrödinger operators with dynamically defined potentials
• Density of positive Lyapunov exponents for SL(2,R) cocycles

Related

• On the almost reducibility conjecture
• KAM, Lyapunov exponents, and the Spectral Dichotomy for typical one-frequency Schrodinger operators
• Quantitative Reducibility of C^k Quasi-Periodic Cocycles
• Reducibility without KAM
• Reducibility of finitely differentiable quasi-periodic cocycles and its spectral applications