Linear Flows on Vector Bundles

Abstract

In this chapter, the exponential growth behavior of linear flows on vector bundles is analyzed. Linear flows (also called linear extensions) are a far-reaching generalization of time-invariant linear differential equations; roughly, they may be viewed as linear differential equations, where the coefficients are determined by a dynamical system on a metric base space B. Section 5.1 describes some examples for these systems (in particular, linearized control systems), introduces the relevant notions, and presents the main results. Section 5.2 constructs the decomposition into invariant subbundles corresponding to the finest Morse decomposition of the induced flow on the projective bundle. It turns out that this coincides with the finest decomposition into exponentially separated subbundles. Furthermore, a uniformity lemma for exponential growth rates is proved, which is needed for the construction of stable manifolds. Section 5.3 studies the associated Morse spectrum, and Section 5.4 uses ergodic theory to study the boundary points of the Morse spectrum. In Section 5.5, the Morse spectrum is related to the Sacker -Sell (or dichotomy) spectrum and the Oseledets spectrum. Section 5.6 proves an abstract version of an invariant manifold theorem due to Bronstein-Chernii for nonlinear (Lipschitzean) perturbations of a linear flow. The proof is based on Hadamard’s graph transform method. Finally, for subbundles with negative Lyapunov exponents one obtains corresponding stable manifolds.