EnglishSingular hyperbolicity and homoclinic tangencies of 3-dimensional flows
Post-edited
Auteurs : Crovisier, Sylvain (Auteur de la Conférence)
CIRM (Editeur )
37C10 - Vector fields, flows, ordinary differential equations
37C29 - Homoclinic and heteroclinic orbits
37Dxx - Dynamical systems with hyperbolic behavior
37F15 - Expanding maps; hyperbolicity; structural stability
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
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Informations sur la Vidéo
Langue : AnglaisDate de publication : 02/03/2017 Date de captation : 21/02/2017 Sous collection : Research talks arXiv category : Dynamical Systems Domaine : Dynamical Systems & ODE Format : MP4 (.mp4) - HD Durée : 00:53:31 Audience : Researchers Download : https://videos.cirm-math.fr/2017-02-21_Crovisier.mp4 
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Informations sur la Rencontre
Nom de la rencontre : Non uniformly hyperbolic dynamical systems. Coupling and renewal theory / Systèmes dynamiques non uniformement et partiellement hyperboliques. Couplage, renouvellementOrganisateurs de la rencontre : 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
Données de citation
DOI : 10.24350/CIRM.V.19128603Citer cette vidéo: 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 |
Bibliographie
- 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
- Crovisier, S., & Yang, D. (2017). Homoclinic tangencies and singular hyperbolicity for three-dimensional vector fields.
- Guckenheimer, J., & Williams, R.F. (1979). Structural stability of Lorenz attractors. Publications Mathématiques, 50, 59-72 - http://dx.doi.org/10.1007/BF02684769
- 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
- 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
