1 - The Concept of Nonuniform Hyperbolicity

Summary

In this chapter we consider sequences of linear maps in the Euclidean space and we introduce the principal notions of Lyapunov exponents, Lyapunov–Perron regularity, normal bases, and so on. These concepts are used in a variety of settings of which the main one is the study of linearizations of a dynamical system along its orbits. Thus a sequence of linear maps can be thought of as the sequence of derivatives (differentials) of a smooth map along an orbit.

We stress that in the situations we consider there are no preferred coordinate systems. Accordingly, even though we often use matrix representations of linear maps, we only study properties that are independent of certain classes of coordinate changes. The most narrow class is that of orthogonal coordinate changes; in the smooth situation, this corresponds to fixing a Riemannian metric in the phase space. A broader class includes coordinate changes uniformly bounded from above and below; in the case of a smooth system on a compact space, this corresponds to an arbitrary choice of a smooth coordinate atlas.

As it turns out, of greatest importance for the theory developed in this book is still a broader class of tempered coordinate changes. This reflects the primary role that exponential behavior plays in our considerations. A tempered change allows arbitrarily large distortions if these distortions change with time with a subexponential rate. Thus it preserves not only the exponential character of the asymptotic behavior but also the actual rates of expansion and contraction in various directions.