Almost reducibility of analytic quasi-periodic cocycles

Claire Chavaudret (1)

1 IMJ - Institut de Mathématiques de Jussieu

Let $G\subset GL(n,\mathbb{C})$ a classical Lie group, $\mathcal{G}$ the Lie algebra associated to $G$, $\omega\in \mathbb{R}^d$ a diophantine vector, $A\in \mathcal{G}$ and a map $F\in C^\omega_r(\mathbb{T}^d,\mathcal{G})$ which is analytic on a neighbourhood of the torus of radius $r\leq \frac{1}{2}$, and $r'\in ]0,r[$. There exists $\epsilon$ depending only on $n,d, A, r-r'$ and on the diophantine class of $\omega$ such that if $\mid F\mid_r \leq \epsilon$, then the quasi-periodic cocycle generated by $A+F$ is almost reducible in $C^\omega_{r'}(2\mathbb{T}^d,G)$. If $G$ is a complex Lie group or $n=2$, almost reducibility holds in $C^\omega_{r'}(\mathbb{T}^d,G)$ and reducible cocycles are dense near constant cocycles in a real analytic topology.

Mots clés: quasi-periodic; cocycles; skew-product; reducibility; almost reducibility

Domaines: Systèmes dynamiques [math.DS]

Claire Chavaudret. Almost reducibility of analytic quasi-periodic cocycles. 2009. ⟨hal-00442730v1⟩