Smoothness of spectral subbundles and reducibility of quasi-periodic linear differential systems

In this paper we study linear differential systems (1) x′ = A ̃(θ + ωt)x, where A ̃(θ) is an (n × n) matrix-valued function defined on the k-torus T^k and (θ, t) → θ + ωt is a given irrational twist flow on T^k. First, we show that if A ε{lunate} C^N(T^k), where N ε{lunate} {0, 1, 2,...; ∞; ω}, then the spectral subbundles are of class C^N on T^k. Next we assume that à is sufficiently smooth on T^k and ω satisfies a suitable "small divisors" inequality. We show that if (1) satisfies the "full spectrum" assumption, then there is a quasi-periodic linear change of variables x = P(t)y that transforms (1) to a constant coefficient system y′ =By. Finally, we study the case where the matrix A ̃(θ + ωt) in (1) is the Jacobian matrix of a nonlinear vector field f{hook}(x) evaluated along a quasi-periodic solution x = φ(t) of (2) x′ = f{hook}(x). We give sufficient conditions in terms of smoothness and small divisors inequalities in order that there is a coordinate system (z, θ{symbol}) defined in the vicinity of Ω = H(φ), the hull of φ, so that the linearized system (1) can be represented in the form z′ = Dz, θ{symbol}′ = ω, where D is a constant matrix. Our results represent substantial improvements over known methods because we do not require that à be "close to" a constant coefficient system.