The basic aim of this book is to present a simple and accessible account of some of the most basic ideas in the theory of non-uniformly hyperbolic diffeomorphisms, or more colloquially, ‘Pesin theory’.

Part I consists of four chapters which contain basic material on the Oseledec theorem, the Ruelle-Pesin inequality, and the Pesin set. There is then a brief ‘interlude’ to mention some topical examples and to draw some motivation from the uniformly hyperbolic (or ‘Axiom A’) case. Then, Part II contains contains three chapters dealing with applications of this theory to periodic points, homoclinic points, and stable manifold theory.

In the course of the text I tried to bring out the following two themes

1. (i) Generality. Ultimately we want to arrive at a theory applicable to any smooth diffeomorphism of a compact surface (providing it has non-zero topological entropy);

2. (ii) The rôle of measure theory. In applying the theory it is remarkable how often invariant measures play a crucial role in situations where the hypothesis and conclusion are purely topological. In some sense, the Poincaré recurrence of invariant measures seems to compensate for the absence of the compactness often take for granted in uniformly hyperbolic systems.

This text is based on a short series of lectures I gave in the Centro de Matematica do INIC na Universidade do Porto between March and June 1989. These lectures were intended to give a basic introduction to some of the simpler and more accessible aspects of the theory (both for the benefit of the audience and myself). My choice of presentation was chiefly influenced by the more topologically oriented approaches in the work of Anatole Katok and Sheldon Newhouse.