ABSTRACT
In this chapter, systems with hyperbolic chain recurrent sets are shown to be Rstable. Certain systems are also shown to be structurally stable: (i) Anosov systems, (ii) Morse-Smale systems, and (iii) systems with a hyperbolic chain recurrent set which also satisfy the transversality condition. Although we have given examples of such systems, we have not discussed the prevalence of such systems. At one time it was hoped that any system could be approximated by a structurally stable system. By now, many counter examples have been constructed. For these examples, there is a whole open set of systems that are not structurally stable or even Ω-stable. However, it is possible to prove that there are certain properties which are generic in the sense of Baire category, i.e., any system can be approximated by another for which these properties are true and the condition is open at least in a weak sense. This chapter considers several of the basic generic properties. It also gives a counter-example to the density of structurally stable systems. The proofs of the genericity use methods from transversality theory, so a section develops these ideas.
