Resumen de Reducibility in a Certain Matrix Lie Algebra for Smooth Linear Quasi-periodic System
In this paper we consider the linear quasi-periodic system θ˙ = ω, x˙ = (A + Q(θ ))x, where (x,θ) ∈ Rn × Td , A ∈ g is a n × n constant matrix with different eigenvalues, g is a matrix Lie subalgebra of gl(n, R), ω = ξω¯ ∈ Rd with ξ ∈ O := [ 1 2 , 3 2 ]. Letting s0 = (d + 1)/2 and β = 6n2 + 6τ − 2, we prove that if Q : Td → g belonging to Sobolev spaces Hs+β with each fixed s ≥ s0 is sufficiently small in given Hs0+β norm and ω¯ satisfies Diophantine condition, then there exists a Cantor set E ⊂ O with almost full Lebesgue measure such that for any ξ ∈ E, there exists a quasi-periodic transformation of the form θ = θ, x = eP(θ ) y with P(θ ) ∈ Hs, which reduces above system into a constant systemθ˙ = ω, y˙ = A∗ y where A∗ ∈ 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 (Hs0+β-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
