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

Abstract

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 C N ( T k ), where N ϵ {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 ƒ(x) evaluated along a quasi-periodic solution x = φ( t) of (2) x′ = ƒ(x). We give sufficient conditions in terms of smoothness and small divisors inequalities in order that there is a coordinate system ( z, ϑ) 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′ = 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.