Lyapunov Sequences

Abstract

In this chapter we show how the existence of a strong tempered dichotomy follows from the existence of a strict quadratic Lyapunov sequence. Starting with the simpler case of a tempered dichotomy, we show in this case that the notion can be completely characterized in terms of the existence of a strict quadratic Lyapunov sequence. This includes explicitly constructing such a sequence for any tempered dichotomy. The chapter can be considered as a version for a single trajectory of corresponding results for cocycles over a measure-preserving transformation. Unsurprisingly, the powerful tools of ergodic theory allow that the corresponding hypotheses in the notion of a Lyapunov function are weaker in the context of ergodic theory.