Full Measure Reducibility for Generic One-parameter Family of Quasi-periodic Linear Systems

Abstract

Let ${C^\omega(\Lambda, gl(m,\,\mathbb{C}))}$ be the set of m × m matrices A(λ) depending analytically on a parameter λ in a closed interval ${\Lambda \subset \mathbb{R}}$. Consider one-parameter families of quasi-periodic linear differential equations: ${\dot{X} = (A(\lambda) + g(\omega_{1}t,\ldots, \omega_{r}t,\lambda))X}$, where ${A\in C^\omega(\Lambda, gl(m,\,\mathbb{C})),g}$ is analytic and sufficiently small. We prove that there is an open and dense set ${\mathcal A}$ in ${C^\omega(\Lambda, gl(m,\,\mathbb{C}))}$, such that for each ${A(\lambda) \in \mathcal{A}}$ the equation can be reduced to an equation with constant coefficients by a quasi-periodic linear transformation for almost all ${\lambda \in \Lambda}$ in Lebesgue measure sense provided that g is sufficiently small. The result gives an affirmative answer to a conjecture of Eliasson (In: Proceeding of Symposia in Pure Mathematics).