ABSTRACT
In this chapter, we return to consider examples with complicated invariant sets. We introduce the idea of a hyperbolic invariant set and show that not only periodic orbits can have stable and unstable manifolds, but that a hyperbolic invariant set also has a family of stable manifolds. We give a number of different types of examples which are hyperbolic and give a method to show that the map is topologically transitive on the invariant set. One very important type of example arises from the intersection of stable and unstable manifolds of a saddle periodic orbit. This gives rise to an invariant set called a Smale horseshoe. It is very similar to the invariant set which we found for the quadratic map on the real line. Another important type of hyperbolic invariant set occurs where all nearby orbits tend toward the invariant set. Such an invariant set is called an attractor (with further conditions added). Thus, an attractor is like a periodic sink where the invariant set itself is more complicated topologically. The final class of examples we consider are those with only a finite number of periodic points, called Morse – Smale systems. For these systems we make a connection with the Lefschetz theory through the Morse – Smale inequalities.
