Reducibility in a Certain Matrix Lie Algebra for Smooth Linear Quasi-periodic System

Abstract

In this paper we consider the linear quasi-periodic system

$$\begin{aligned} \left\{ \begin{array}{l} {\dot{\theta }}=\omega ,\\ \dot{x}=(A+Q(\theta ))x,\\ \end{array} \right. \end{aligned}$$

where $(x,\theta )\in \mathbb {R}^n\times \mathbb {T}^d$, $A\in g$ is a $n\times n$ constant matrix with different eigenvalues, g is a matrix Lie subalgebra of $gl(n,\mathbb {R})$, $\omega =\xi \bar{\omega }\in \mathbb {R}^d$ with $\xi \in \mathcal {O}:=[\frac{1}{2},\frac{3}{2}].$ Letting $s_0=(d+1)/2$ and $\beta =6n^2+6\tau -2$, we prove that if $Q:\mathbb {T}^d\rightarrow g$ belonging to Sobolev spaces $H^{s+\beta }$ with each fixed $s \ge s_0$ is sufficiently small in given $H^{s_0+\beta }$ norm and ${\bar{\omega }}$ satisfies Diophantine condition, then there exists a Cantor set ${\mathcal {E}}\subset \mathcal {O}$ with almost full Lebesgue measure such that for any $\xi \in {\mathcal {E}}$, there exists a quasi-periodic transformation of the form $\theta =\theta$, $x = e^{P(\theta )}y$ with $P(\theta )\in H^s$, which reduces above system into a constant system ${\dot{\theta }}=\omega ,$ $\dot{y}=A_* y$ where $A_*\in g$ is a constant matrix close to A. Different from classical smooth results, our result requires smallness conditions only on a fixed low Sobolev norm ($H^{s_0+\beta }$-norm) of the first perturbation. It is worth mentioning that our system does not need second Melnikov’s condition explicitly. As an application, we apply our results to smooth quasi-periodic Schrödinger equations to study the Lyapunov stability of the equilibrium and the existenc of quasi-periodic solutions. The result can be regarded as the generalization of the stability result in [37] to the smooth category.