Almost reducibility for finitely differentiable SL(2,R)-valued quasi-periodic cocycles

TL;DR

This paper proves that finitely differentiable SL(2,R)-valued quasi-periodic cocycles close to constant are almost reducible, extending Eliasson's theorem from the analytic to the differentiable setting.

Contribution

It extends Eliasson's almost reducibility theorem for Schrödinger cocycles from the analytic case to finitely differentiable functions.

Findings

Almost reducibility holds for sufficiently close cocycles in the differentiable topology.

Reducibility occurs when the fibered rotation number is diophantine or rational.

The method involves analytic approximation with a controlled loss of differentiability.

Abstract

Quasi-periodic cocycles with a diophantine frequency and with values in SL(2,R) are shown to be almost reducible as long as they are close enough to a constant, in the topology of k times differentiable functions, with k great enough. Almost reducibility is obtained by analytic approximation after a loss of differentiability which only depends on the frequency and on the constant part. As in the analytic case, if their fibered rotation number is diophantine or rational with respect to the frequency, such cocycles are in fact reducible. This extends Eliasson's theorem on Schr\"odinger cocycles to the differentiable case.