Hyperbolic sets,transversal homoclinic trajectories,and symbolic dynamics for C1-maps in banach spaces

In an earlier paper we generalized the notion of a hyperbolic set and proved that the Shadowing Lemma remains valid, for C¹-maps which need not be invertible. Here we establish the existence of (generalized) hyperbolic structures along transversal homoclinic trajectories of C¹-maps. The hyperbolic structure and shadowing are then used to give a new proof of a result due to Hale and Lin (and ilnikov) on symbolic dynamics forall trajectories sufficiently close to a transversal homoclinic trajectory. The result is applied to a Poincaré map without continuous inverse, which is associated with a periodic orbit of an autonomous differential delay equation.