Lyapunov functions for strong exponential dichotomies

We give a complete characterization of a general type of exponential dichotomies in terms of Lyapunov functions, both for discrete and continuous time. This includes constructing explicitly quadratic Lyapunov functions for each exponential dichotomy. We consider the general cases of nonautonomous dynamics, nonuniform exponential dichotomies and, motivated by ergodic theory, strong exponential dichotomies, in the sense that there exist simultaneously lower and upper contraction and expansion bounds. As a nontrivial application, we establish the persistence of the asymptotic stability of a strong nonuniform exponential dichotomy under sufficiently small linear perturbations.