Structural Stability and Hyperbolicity

In this chapter we want to analyze which one-dimensional systems are structurally stable. In Chapter I this question was quite easy to answer: a circle diffeomorphism is structurally stable if and only if all periodic points of f are hyperbolic. Moreover structurally stable diffeomorphisms form an open and dense set. (These statements were shown in Exercise I.4.1.) For non-invertible maps the situation is much more complicated and partly unknown. The concept of hyperbolicity of some infinite compact set will play an essential role in this discussion. As we will see in this chapter non-invertible one-dimensional dynamical systems have many infinite hyperbolic sets whereas circle diffeomorphisms have none.