Appears in collections : Non uniformly hyperbolic dynamical systems. Coupling and renewal theory / Systèmes dynamiques non uniformement et partiellement hyperboliques. Couplage, renouvellement, Exposés de recherche
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.
By Mikhail Lyubich
By Daniel Monclair
