Hyperbolic sets, transversal homoclinic trajectories, and symbolic dynamics for \(C^ 1\)-maps in Banach spaces (Q915229)

In earlier work, the authors generalized the notion of a hyperbolic set for non-invertible maps, f: \(U\to L\) where U is an open subset of a Banach space L. They also showed that the Shadowing Lemma remains valid for \(C^ 1\) maps which need not be invertible. In this paper, the authors prove the existence of generalized hyperbolic structures along transversal homoclinic trajectories of \(C^ 1\) maps. Using this hyperbolic structure and their extension of the Shadowing Lemma, they present a new proof of a result (originally due to Hale, Lin, and Šitnikov) on symbolic dynamics for all trajectories sufficiently close to a transversal homoclinic trajectory. They then apply this result to a Poincaré map without a continuous inverse, a map which is associated with a periodic orbit of an autonomous differential delay equation.