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Post-edited Singular hyperbolicity and homoclinic tangencies of 3-dimensional flows

Auteurs : Crovisier, Sylvain (Auteur de la Conférence)
CIRM (Editeur )

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dynamics of surface diffeomorphisms dynamics of 3-dimensional vector fields singular hyperbolicity dynamics far from homoclinic tangencies sectional flow uniform contraction

Résumé : The notion of singular hyperbolicity for vector fields has been introduced by Morales, Pacifico and Pujals in order to extend the classical uniform hyperbolicity and include the presence of singularities. This covers the Lorenz attractor. I will present a joint work with Dawei Yang which proves a dichotomy in the space of three-dimensional $C^{1}$-vector fields, conjectured by J. Palis: every three-dimensional vector field can be $C^{1}$-approximated by one which is singular hyperbolic or by one which exhibits a homoclinic tangency.

Codes MSC :
37C10 - Vector fields, flows, ordinary differential equations
37C29 - Homoclinic and heteroclinic orbits
37Dxx - Dynamical systems with hyperbolic behavior
37F15 - Expanding maps; hyperbolicity; structural stability

    Informations sur la Vidéo

    Réalisateur : Hennenfent, Guillaume
    Langue : Anglais
    Date de publication : 02/03/2017
    Date de captation : 21/02/2017
    Collection : Research talks
    Format : MP4 (.mp4) - HD
    Durée : 00:53:31
    Domaine : Dynamical Systems & ODE
    Audience : Chercheurs ; Doctorants , Post - Doctorants
    Download : http://videos.cirm-math.fr/2017-02-21_Crovisier.mp4

Informations sur la rencontre

Nom du congrès : Non uniformly hyperbolic dynamical systems. Coupling and renewal theory / Systèmes dynamiques non uniformement et partiellement hyperboliques. Couplage, renouvellement
Organisteurs Congrès : Troubetzkoy, Serge ; Vaienti, Sandro
Dates : 20/02/17 - 24/02/17
Année de la rencontre : 2017
URL Congrès : http://conferences.cirm-math.fr/1714.html

Citation Data

DOI : 10.24350/CIRM.V.19128603
Cite this video as: Crovisier, Sylvain (2017). Singular hyperbolicity and homoclinic tangencies of 3-dimensional flows. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19128603
URI : http://dx.doi.org/10.24350/CIRM.V.19128603

Voir aussi


  1. Afrajmovich, V.S., Bykov, V.V., & Shil'nikov, L.P. (1977). On the origin and structure of the Lorenz attractor. Soviet Physics. Doklady, 22, 253-255 - https://zbmath.org/?q=an:03706105

  2. Crovisier, S., & Yang, D. (2017). Homoclinic tangencies and singular hyperbolicity for three-dimensional vector fields. <arXiv:1702.05994> - https://arxiv.org/abs/1702.05994

  3. Guckenheimer, J., & Williams, R.F. (1979). Structural stability of Lorenz attractors. Publications Mathématiques, 50, 59-72 - http://dx.doi.org/10.1007/BF02684769

  4. Morales, C.A., Pacifico, M.J., & Pujals, E.R. (2004). Robust transitive singular sets for 3-flows are partially hyperbolic attractors or repellers. Annals of Mathematics. Second Series, 160(2), 375-432 - http://dx.doi.org/10.4007/annals.2004.160.375

  5. Pujals, E.R., & Sambarino, M. (2000). Homoclinic tangencies and hyperbolicity for surface diffeomorphisms. Annals of Mathematics. Second Series, 151(3), 961-1023 - http://dx.doi.org/10.2307/121127