Existence of dichotomies and invariant splittings for linear differential systems, III

This paper is primarily concerned with linear time-varying ordinary differential equations. Sufficient conditions are given for the existence of a "trichotomy," i.e., a continuous decomposition of Rⁿ into stable, unstable and neutral subspaces. For constant coefficients it reduces to the usual (Jordan) decomposition of Rⁿ into subspaces corresponding to eigenvalues with negative, positive, and zero real parts, respectively, but only in the case in which the eigenvalues with zero real parts occur with simple elementary divisors. The conditions are related to those used by Favard in his study of almost periodic equations. The problem is treated in the unified setting of a skew-product dynamical system and the results apply to discrete systems including those generated by diffeomorphisms of manifolds. In the continuous case, sufficient conditions are given for a flow on a compact manifold to be an Anosov flow.