Spectral Properties of Pullback Operators on Vector Bundles of a Dynamical System

The spectrum of the Koopman operator has been shown to encode many important statistical and geometric properties of a dynamical system. In this work, we consider induced linear operators acting on the space of sections of the state space's tangent, cotangent, and tensor bundles. We first demonstrate how these operators are natural generalizations of Koopman operators acting on functions. We then draw connections between the various operators' spectra and characterize the algebraic and differential topological properties of their spectra. We describe the discrete spectrum of these operators for linear dynamical systems and derive spectral expansions for linear vector fields. We define the notion of an ``eigendistribution,"" provide conditions for an eigendistribution to be integrable, and demonstrate how to recover the foliations arising from their integral manifolds. Last, we demonstrate that the characteristic Lyapunov exponents of a uniformly hyperbolic dynamical system are in the spectrum of the induced operators on sections of the tangent or cotangent bundle. We conclude with an application to differential geometry where the well-known fact that the flows of commuting vector fields commute is generalized, and we recover the original statement as a particular case of our result. We also apply our results to recover the Lyapunov exponents and the stable/unstable foliations of Arnold's cat map via the spectrum of the induced operator on sections of the tangent bundle.