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

  • Heinrich Steinlein
  • Hans-Otto Walther

DOI: 10.1007/BF01048949

Cite this article as:
Steinlein, H. & Walther, HO. J Dyn Diff Equat (1990) 2: 325. doi:10.1007/BF01048949


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 šilnikov) 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.

Key words

Hyperbolic set noninvertible C1-map shadowing transversal homoclinic trajectory symbolic dynamics Poincaré map autonomous differential delay equation 

AMS (MOS) classifications

58F15 34C35 

Copyright information

© Plenum Publishing Corporation 1990

Authors and Affiliations

  • Heinrich Steinlein
    • 1
  • Hans-Otto Walther
    • 1
  1. 1.Mathematisches InstitutUniversitÄt MünchenMünchen 2West Germany

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