From one-sided dichotomies to two-sided dichotomies

Abstract

For a general nonautonomous dynamics on a Banach space, we give a necessary and sufficient condition so that the existence of one-sided exponential dichotomies on the past and on the future gives rise to a two-sided exponential dichotomy. The condition is that the stable space of the future at the origin and the unstable space of the past at the origin generate the whole space. We consider the general cases of a noninvertible dynamics as well as of a nonuniform exponential dichotomy and a strong nonuniform exponential dichotomy (for the latter, besides the requirements for a nonuniform exponential dichotomy we need to have a minimal contraction and a maximal expansion). Both notions are ubiquitous in ergodic theory. Our approach consists in reducing the study of the dynamics to one with uniform exponential behavior with respect to a family of norms and then using the characterization of uniform hyperbolicity in terms of an admissibility property in order to show that the dynamics admits a two-sided exponential dichotomy. As an application, we give a complete characterization of the set of Lyapunov exponents of a Lyapunov regular dynamics, in an analogous manner to that in the Sacker--Sell theory.