1 ERGODIC THEORY of DIFFERENTIABLE DYNAMICAL SYSTEMS

These notes are about the dynamics of systems with hyperbolic properties. The setting for the first half consists of a pair (f, μ), where f is a diffeomorphism of a Riemannian manifold and μ is an f -invariant Borel probability measure. After a brief review of abstract ergodic theory, Lyapunov exponents are introduced, and families of stable and unstable manifolds are constructed. Some relations between metric entropy, Lyapunov exponents and Hausdorff dimension are discussed. In the second half we address the following question: given a differentiable mapping, what are its natural invariant measures? We examine the relationship between the expanding properties of a map and its invariant measures in the Lebesgue measure class. These ideas are then applied to the construction of Sinai-RuelleBowen measures for Axiom A attractors. The nonuniform case is discussed briefly, but its details are beyond the scope of these notes. I have aimed these notes at readers who have a basic knowledge of dynamics but who are not experts in the ergodic theory of hyperbolic systems. To cover the material mentioned above in 40-50 pages, some choices had to be made. I wanted very much to give the reader some feeling for the flavor of the subject, even if that meant focusing on fewer ideas. I have not hesitated to include examples, informal discussions, and some of my favorite proofs. I did not try to mention all related results. For survey articles on similar topics see [ER] or [S5].