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Singular hyperbolicity and homoclinic tangencies of 3-dimensional flows
Post-edited
Authors : Crovisier, Sylvain (Author of the conference)
CIRM (Publisher )
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 (Publisher )
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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 |
Abstract :
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.
MSC Codes : 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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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 02/03/2017 Conference Date : 21/02/2017 Subseries : Research talks arXiv category : Dynamical Systems Mathematical Area(s) : Dynamical Systems & ODE Format : MP4 (.mp4) - HD Video Time : 00:53:31 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2017-02-21_Crovisier.mp4 
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Information on the Event
Event Title : Non uniformly hyperbolic dynamical systems. Coupling and renewal theory / Systèmes dynamiques non uniformement et partiellement hyperboliques. Couplage, renouvellementEvent Organizers : Troubetzkoy, Serge ; Vaienti, Sandro Dates : 20/02/17 - 24/02/17 Event Year : 2017 Event URL : http://conferences.cirm-math.fr/1714.html
Citation Data
DOI : 10.24350/CIRM.V.19128603Cite 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 |
See Also
- [Multi angle] Fast-Slow partially hyperbolic systems: an example / Author of the conference Liverani, Carlangelo.
- [Multi angle] Mixing and rates of mixing for infinite measure flows / Author of the conference Melbourne, Ian.
- [Multi angle] Central limit theorems for circle packings / Author of the conference Pollicott, Mark.
- [Multi angle] Generic properties of the geodesic flow in nonpositive curvature / Author of the conference Coudène, Yves.
Bibliography
- 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
