Reducibility of cocycles under a Brjuno-R\"ussmann arithmetical condition

The arithmetics of the frequency and of the rotation number play a fundamental role in the study of reducibility of analytic quasi-periodic cocycles which are sufficiently close to a constant. In this paper we show how to generalize previous works by L.H.Eliasson which deal with the diophantine case so as to implement a Brjuno-Russmann arithmetical condition both on the frequency and on the rotation number. Our approach adapts the Poschel-Russmann KAM method, which was previously used in the problem of linearization of vector fields, to the problem of reducing cocycles

Normal form of holomorphic vector fields with an invariant torus under Brjuno's A condition

We consider the holomorphic normalization problem for a holomorphic vector field in the neighborhood of the product of a fixed point and an invariant torus. Supposing that the vector field is a perturbation of a linear part around the fixed point and of a rotation on the invariant torus (the unperturbed vector field is called the quasi-linear part of the perturbed one), it was shown by J.Aurouet that the system is holomorphically linearizable if there are no exact resonances in the quasi-linear part and if the quasi-linear part satisfies to Brjuno's arithmetical condition. In the presence of exact resonances, a conjecture by Brjuno states that the system will still be holomorphically conjugated to a normal form under the same arithmetical condition and a strong algebraic condition on the formal normal form. This article proves this conjecture.Comment: Annales de l'Institut Fourier, Institut Fourier, 201

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

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

Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles

This paper is about almost reducibility of quasi-periodic cocycles with a diophantine frequency which are sufficiently close to a constant. Generalizing previous works by L.H.Eliasson, we show a strong version of almost reducibility for analytic and Gevrey cocycles, that is to say, almost reducibility where the change of variables is in an analytic or Gevrey class which is independent of how close to a constant the initial cocycle is conjugated. This implies a result of density, or quasi-density, of reducible cocycles near a constant. Some algebraic structure can also be preserved, by doubling the period if needed

Almost reducibility of quasiperiodic sl(2, R)-cocycles in ultradifferentiable classes

Given a quasiperiodic cocycle in sl(2, R) sufficiently close to a constant, we prove that it is almost-reducible in ultradifferentiable class under an adapted arithmetic condition on the frequency vector. We also give a corollary on the H{\"o}lder regularity of the Lyapunov exponent