Spectral theory for linearizations of dynamical systems

Abstract

The description of the asymptotic growth rates for the solution trajectories of a system of differential equations is a central problem in the qualitative theory of dynamical systems. These growth rates are described by the eigenvalues of the Jacobian matrix for trajectories near a critical point, by the Floquet exponents for trajectories close to a periodic orbit and, generally, by the Lyapunov characteristic exponents in a neighborhood of an arbitrary trajectory. In the examples, the asymptotic behavior of the solution trajectories is given by the association of a “spectrum,” and this is a recurrent theme in the method of linearization. Sacker and Sell [IO] recently introduced a spectral theory for differential systems of quite general type: the (linear) skew-product flows (see Section 3). Their work is in the spirit of the classical linearization theory and is closely tied to the idea of the Lyapunov exponents. We prefer to use the word “spectrum” literally, identifying it with the spectrum of an operator on a Banach space. In general, we consider a one-parameter group 4’ of homeomorphisms of a compact metric space M which, for example, may be the solution flow of a differential equation on a smooth manifold. Second, we assume that a continuous vector bundle (E, M) is defined, i.e., a locally trivial continuous map n: E -+ M with linear fibers which are isomorphic to either real n-dimensional space R” or complex n-dimensional space C”. In case #’ is the flow of a vector field X on M, this vector bundle could be the tangent bundle TM. Finally, we assume that there is group of homeomorphisms @’ on E with n# = 4% and such that @i: E, --t E,,c,, is a linear isomorphism for each x E M. For instance, in case 4’ is smooth, Cp’ could be the tangent flow on TM given by the derivative of 4’. Using the compactness of M we define a Finsler on E which continuously assigns a norm, denoted / I, to each fiber E,. With this assignment, the space r(E) of continuous sections of E (functions q: M + E such that 7~ o q(x) = x) forms a Banach space with norm IIqll= supXEM u x 155 1 ( )I. The topology of T(E) is