Morse-Smale systems and horseshoes  for three dimensional singular flows

Gan, Shaobo; Yang, Dawei

doi:10.24033/asens.2351

Morse-Smale systems and horseshoes for three dimensional singular flows
[Systèmes Morse-Smale et fers-à-cheval pour les flots singuliers en dimension 3]

Gan, Shaobo ; Yang, Dawei

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 1, pp. 39-112.

Résumé
Abstract

Nous montrons que tout champ de vecteurs en dimension trois peut être accumulé en topologie $C^{1}$ ou bien par un champ Morse-Smale, ou bien par un champ possédant une intersection homocline transverse associée à une orbite périodique hyperbolique. Contrairement au cas des difféomorphismes [14], la principale difficulté ici consiste à traiter les singularités. Nous progressons également en direction d'une autre conjecture de Palis.

We prove that every three-dimensional vector field can be $C^{1}$ accumulated by Morse-Smale ones, or by ones with a transverse homoclinic intersection of some hyperbolic periodic orbit. In contrast to the case of diffeomorphisms [14], the main difficulty here is that we need to deal with singularities. We also make progress on another conjecture related to Palis in this paper.

DOI : 10.24033/asens.2351

Classification : 37C10, 37C20, 37C29, 37D15, 37D30.
Keywords: Morse-Smale system, horseshoe, vector field, singularity.
Mot clés : Système de Morse-Smale, fer-à-cheval, champ de vecteurs, singularité.

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     title = {Morse-Smale systems and horseshoes  for three dimensional singular flows},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {39--112},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
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Gan, Shaobo; Yang, Dawei. Morse-Smale systems and horseshoes  for three dimensional singular flows. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 1, pp. 39-112. doi : 10.24033/asens.2351. http://www.numdam.org/articles/10.24033/asens.2351/

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