Almost reducibility and absolute continuity I

By Artur Avila. 2010

97 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/88283479fe8b88fb68c8b74cac38726b1d37b73d

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Analytic Quasi-Perodic Cocycles with Singularities and the Lyapunov Exponent of Extended Harper’s Model (2010, background): “For almost periodic Schrödinger operators with analytic potentials the almost reducibility conjecture (ARC) [2, 18, 19, 20] claims a one to one correspondence between regularity of the Schrödinger cocycle and purely absolutely continuous spectrum.”
Schrödinger operators with dynamically defined potentials (2014, background): “By showing that subcriticality implies almost-reducibility (which for some time was referred to as the ‘almost reducibility conjecture’ (see [16]), at least until it was proved in [4, 6]), Avila was then able to show that the subcritical regime in fact must be purely absolutely continuous.”