On the Almost Reducibility Conjecture

Avila’s Almost Reducibility Conjecture (ARC) is a powerful statement linking purely analytic and dynamical properties of analytic one-frequency cocycles. It is also a fundamental tool in the study of spectral theory of analytic one-frequency Schrödinger operators, with many striking consequences, al...

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Published inGeometric and functional analysis Vol. 34; no. 1; pp. 32 - 59
Main Author Ge, Lingrui
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.02.2024
Springer Nature B.V
Subjects
Analysis
Frequency analysis
Mathematical analysis
Mathematics
Mathematics and Statistics
Operators (mathematics)
Polynomials
Spectral theory
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Summary:Avila’s Almost Reducibility Conjecture (ARC) is a powerful statement linking purely analytic and dynamical properties of analytic one-frequency cocycles. It is also a fundamental tool in the study of spectral theory of analytic one-frequency Schrödinger operators, with many striking consequences, allowing to give a detailed characterization of the subcritical region. Here we give a proof, completely different from Avila’s, for the important case of Schrödinger cocycles with trigonometric polynomial potentials and non-exponentially approximated frequencies, allowing, in particular, to obtain all the desired spectral consequences in this case.
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ISSN:1016-443X
1420-8970
DOI:10.1007/s00039-024-00671-0