ABSTRACT
In this chapter, the authors look at the Smale’s horseshoe maps, concentrating on the general properties of horseshoes (in other words, concentrating on general dynamical features associated with the existence of homoclinic orbits in maps) without reference to the specific facts of the dynamics where it may appear. The word horseshoe will be used a little loosely to indicate either the map (that bends its domain in the form of a horseshoe) or the invariant set. Chaotic attractors of bi-dimensional systems are more subtle sets than ‘simply’ horseshoes. Their relation with horseshoes has yet to be fully elucidated. The following theorem due to Katok shows that there is in fact an important relation between chaotic attractors and horseshoe maps. Hyperbolic invariant sets have a property that is crucial to understand the meaning of numerical experiments done on a horseshoe. This is called the shadowing lemma.
