Hyperbolic sets, transversal homoclinic trajectories, and symbolic dynamics for \(C^ 1\)-maps in Banach spaces
type: Item
instance of: scholarly article
title: Hyperbolic sets, transversal homoclinic trajectories, and symbolic dynamics for \(C^ 1\)-maps in Banach spaces
zbMATH Open document ID: 0702.58051
DOI: 10.1007/BF01048949
author: Heinrich Steinlein
author: Hans-Otto Walther
published in: Journal of Dynamics and Differential Equations
publication date: 1990-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 37D99
Mathematics Subject Classification ID: 37E99
zbMATH DE Number: 4151451
zbMATH Keywords: hyperbolic set
zbMATH Keywords: generalized hyperbolic structures
zbMATH Keywords: transversal homoclinic trajectories
zbMATH Keywords: symbolic dynamics
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01048949
OpenAlex ID: W2083533842
description: scientific article; zbMATH DE number 4151451