A Hartman-Grobman theorem on Banach bundles

Abstract

We provide generalized linearization theorems for skew-product flows on abstract Banach bundles over locally compact metric spaces, allowing for a unified treatment of uniform and nonuniform, exponential and strong exponential dichotomies. This is accomplished by constructing an operator whose fixed points are exactly the conjugacies that are bounded perturbations of the identity, and by giving criteria under which this operator is a contraction. Under stronger assumptions we refine the argument to obtain H\"older regularity of the conjugacy. Finally, we apply our results to recover known results for non-autonomous differential equations and to obtain new linearization results for control-affine systems on manifolds.