Lyapunov Exponents and Regularity

Abstract

This chapter is an introduction to the basic theory of Lyapunov exponents. We first introduce the notions of a Lyapunov exponent and of Lyapunov regularity in terms of the Grobman coefficient. This is part of what is usually called the abstract theory of Lyapunov exponents. We then illustrate the notions with two specific classes of Lyapunov exponents obtained from a linear dynamics. More precisely, we consider linear dynamics with discrete and continuous time, with the study, respectively, of the Lyapunov exponents defined by sequences of invertible matrices and by nonautonomous linear differential equations. These two classes of Lyapunov exponents are the main objects of study in the book. Finally, we introduce the Perron coefficient and we use it to give an alternative characterization of Lyapunov regularity, in terms of the values of the dual Lyapunov exponents used to define the Grobman coefficient.