The Spectral Characterization of Normal Hyperbolicity

In many cases hyperbolicity in dynamical systems can be expressed in terms of the spectrum of some canonically associated linear operator; e.g., the linearization at a fixed point. Such a characterization is known for Anosov diffeomorphisms and flows. We construct a vector bundle map, based on the tensor product, whose spectrum is decisive for detecting the normal hyperbolicity of a flow or diffeomorphism at an invariant manifold. This resolves a problem raised by Hirsch, Pugh and Shub. In the case of flows, our operator admits an infinitesimal formulation, which allows us to prove that normally hyperbolic systems are stable under reparameterization in many cases.