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

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Abstract

In this paper we study linear differential systems (1) x′ = Ã(θ + ωt)x, whereÃ(θ) is an (n × n) matrix-valued function defined on the k-torus Tk and (θ, t) → θ + ωt is a given irrational twist flow on Tk. First, we show that if A ϵ CN(Tk), where N ϵ {0, 1, 2,…; ∞; ω}, then the spectral subbundles are of class CN on Tk. Next we assume that à is sufficiently smooth on Tk 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 Ã(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.

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This research was supported in part by NSF Grant MCS 79-01998. This work was done while the first author was visiting at the University of Minnesota.