Tempered Dichotomies

Abstract

In this chapter we start describing the important relation between nonzero Lyapunov exponents and hyperbolicity. More precisely, we discuss how the existence of a nonzero Lyapunov exponent gives rise to hyperbolicity and how this relates to the theory of regularity. We start with the simpler case of sequences of matrices with a negative Lyapunov exponent, for which the exposition is simpler. We also consider the notion of strong tempered spectrum of a sequence of matrices and we describe all its possible forms. This spectrum can be seen as a nonuniform version of the Sacker–Sell spectrum, which in its turn can be described as a nonautonomous version of the spectrum of a matrix. In the case of a regular sequence of matrices, we show that the strong tempered spectrum is simply the set of values of the Lyapunov exponent. We also describe briefly versions of the former results for continuous time.