ABSTRACT
In this chapter, we take the ideas introduced in the last chapter and make them into a more complete theory. The first section shows that in some sense any map can be decomposed into a map on chain recurrent pieces and a gradient-like map (or flow) between the pieces. This is a very general theorem of Conley and can be proved without using much of the material introduced earlier in this book except the definition of chain recurrent. The next section indicates how the proof of the stable manifold theorem for a fixed point can be modified to prove the case for a hyperbolic invariant set. Using these results, we prove the possibility of shadowing near a hyperbolic invariant set, the Ω-stability of diffeomorphisms with a hyperbolic chain recurrent set, and the structural stability of diffeomorphisms which satisfy a “transversality” condition in addition to having a hyperbolic chain recurrent set. These theorems form the heart of the theory of hyperbolic diffeomorphisms (ones for which the chain recurrent set is hyperbolic) which was articulated by Smale (1967) and carried out in the following years by Smale and other researchers.
