Reducibility of slow quasi-periodic linear systems

Wu, Jian; You, Jiangong

doi:10.1090/S0002-9939-2013-11915-5

Reducibility of slow quasi-periodic linear systems
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by Jian Wu and Jiangong You PDF

Proc. Amer. Math. Soc. 141 (2013), 3147-3155 Request permission

Abstract:

In this note, we prove that the reducibility of analytic quasi-periodic linear systems close to constant is irrelevant to the size of the base frequencies. More precisely, we consider the quasi-periodic linear systems \[ \dot {X} =(A+B(\theta ))X,\quad \dot {\theta }=\lambda ^{-1}\omega \] in $\mathbb {C}^{m},$ where the matrix $A$ is constant and $\omega$ is a fixed Diophantine vector, $\lambda \in \mathbb {R}\backslash \{0\}$. We prove that the system is reducible for typical $A$ if $B(\theta )$ is analytic and sufficiently small (depending on $A, \omega$ but not on $\lambda$).

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