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

Authors:	Hai-Long Her Jiangong You

Institution:	(1) Department of Mathematics, Nanjing University, Nanjing, 210093, China

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). Dedicated to Professor Zhifen Zhang on the occasion of her 80th birthday

Keywords:	Reducibility Quasi-periodic KAM
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