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Singular hyperbolicity and homoclinic tangencies of 3-dimensional flows

By Sylvain Crovisier

Also appears in collection : Non uniformly hyperbolic dynamical systems. Coupling and renewal theory / Systèmes dynamiques non uniformement et partiellement hyperboliques. Couplage, renouvellement

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.

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Citation data

  • DOI 10.24350/CIRM.V.19128603
  • Cite this video Crovisier, Sylvain (21/02/2017). Singular hyperbolicity and homoclinic tangencies of 3-dimensional flows. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19128603
  • URL https://dx.doi.org/10.24350/CIRM.V.19128603

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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. <arXiv:1702.05994> - https://arxiv.org/abs/1702.05994
  • 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

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