Reducibility of quasiperiodic cocycles under a Brjuno-Rüssmann arithmetical condition

Abstract

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-Rüssmann arithmetical condition both on the frequency and on the rotation number. Our approach adapts the Pöschel-Rüssmann KAM method, which was previously used in the problem of linearization of vector fields, to the problem of reducing cocycles.Quasiperiodic cocycles are the fundamental solutions of quasi-periodic linear systems

KAM, α-Gevrey regularity and the α-Bruno-Rüssmann condition

“…Finally, one may consider the problem of reducibility of quasi-periodic cocycles close to constant. In the analytic case, the Bruno-Rüssmann condition is sufficient, as was shown in [CM12]; in the α-Gevrey case, the α-Bruno-Rüssmann condition is sufficient. In fact, this setting is simpler from a technical point of view and our Gevrey estimates are not necessary to obtain such a result; one simply needs to go through the proof of [CM12].…”

“…In the analytic case, the Bruno-Rüssmann condition is sufficient, as was shown in [CM12]; in the α-Gevrey case, the α-Bruno-Rüssmann condition is sufficient. In fact, this setting is simpler from a technical point of view and our Gevrey estimates are not necessary to obtain such a result; one simply needs to go through the proof of [CM12]. A possible explanation for this is that for quasi-periodic cocycles, composition occur in a linear Lie group, thus only estimates for linear composition (product of matrices) are necessary and so everything boils down to good estimates for the product of two functions.…”

Reducibility of three dimensional skew symmetric system with Liouvillean basic frequencies

“…where γ 0 > 0, and is the Brjuno-Rüssmann approximation function as in (9). For the KAM theory under Brjuno-Rüssmann's non-resonant condition, we see [4,20,21,22,27,31,32,33].…”

Analytic Reducibility of Resonant Cocycles to a Normal Form

“…Proof. Since, for all t,g(t) g(t), property (a) follows (6). Property (c) holds by definition ofg, sinceg(t) φ −1 (t) 1/4 for all t 1, and in particular for t = φ(ε), ε ∈ (0, 1/e).…”