Ergodic Properties of Linear Dynamical Systems

Abstract

The Multiplicative Ergodic Theorem give information about the dynamical structure of a cocycle , or a linear skew product flow , over a suitable base space M. In typical applications the base space M is either an attractor; a compact invariant set; or the space of coefficients for a diffeomorphism, a differential equation, or a vector field. This theorem asserts that for every invariant probability measure on M there is a measurable decomposition of the vector bundle over M into invariant measurable subbundles, and that every solution with initial conditions in any of these subbundles has strong Lyapunov exponets. These exponents, or growth rates, depend on the measure , and when is ergodic, they are constant (almost everywhere) on M and form a finite set meas(), the measurable (Millionscikov-Oseledec) spectrum. The main objective in this paper is to study the connection between the measurable spectrum meas() and the dynamical spectrum dyn introduced by Sacker and Sell (1975, 1978, 1980). (Also see Daletskii and Krein (1974), as well as Selgrade (1975). The dynamical spectrum dyn consists of those values R for which the shifted flow fails to have an exponential dichotomy over M. It follows from the Spectral Theorem, Sacker and Sell (1978), that the dynamical spectrum is the finite union of disjoint compact intervals when M is compact and dynamically connected.