Lyapunov functions for strong exponential contractions

Abstract

Our main aim is to give a complete characterization of an exponential contraction in terms of Lyapunov sequences (for discrete time) or Lyapunov functions (for continuous time). In particular, we obtain inverse theorems giving explicitly a quadratic Lyapunov function for each exponential contraction. We consider the general cases of: (1) nonautonomous dynamics, either obtained from a product of linear operators or from a nonautonomous differential equation, respectively for discrete and continuous time; (2) nonuniform exponential contractions, in which the uniform stability is replaced by a nonuniform stability; (3) strong exponential behavior, in the sense that we have simultaneously lower and upper contraction bounds. We emphasize that the two last properties are ubiquitous in the context of ergodic theory and, in particular, for almost all trajectories of any measure-preserving flow, which justifies completely the study of this general type of exponential behavior. As a nontrivial application of our work, we establish the robustness of any strong nonuniform exponential contraction, that is, the persistence of the asymptotic stability of a strong nonuniform exponential contraction under sufficiently small linear perturbations.