Morse-Smale systems and horseshoes  for three dimensional singular flows

Gan, Shaobo; Yang, Dawei

doi:10.24033/asens.2351

Geodesic

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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.

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Abstract (VO)
Résumé

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.

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.

DOI : 10.24033/asens.2351

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

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     pages = {39--112},
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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://geodesic.mathdoc.fr/articles/10.24033/asens.2351/

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