The spectrum of an invariant submanifold

Abstract

This paper is concerned with vector fields on smooth compact manifolds. The exponential growth of solutions of the linearized equations is described by the already well-known Spectral Theorem applied to the induced linear flow on the tangent bundle. The spectrum of the tangent bundle flow is compared to the two secondary spectra obtained by first taking the spectrum of the bundle of tangent spaces to an invariant submanifold and second, the spectrum of an induced flow on an arbitrary complementary bundle to the latter. The relationship among the three spectra is studied and it is shown that whenever these secondary spectra are disjoint then an invariant complementary bundle can be found. The results have implications in the theory of perturbation of invariant manifolds. The problem is studied in the setting of skew-product dynamical systems and the results are applicable to block triangular systems of ordinary differential equations.