Strong regularity

"The strong regularity program was initiated by Jean-Christophe Yoccoz during his first lecture at Collège de France. As explained in the first article of this volume, this program aims to show the abundance of dynamics displaying a non-uniformly hyperbolic attractor. It proposes a topological and combinatorial definition of such mappings using the formalism of puzzle pieces. Their combinatorics enable to deduce the wished analytical properties. In 1997, this method enabled Jean-Christophe Yoccoz to give an alternative proof of the Jakobson theorem: the existence of a set of positive Lebesgue measure of parameters a such that the map x x^2 + a has an attractor which is non-uniformly hyperbolic. This proof is the second article of this volume. In the third article, this method is generalized in dimension 2 by Pierre Berger to show the following theorem. For every C^2-perturbation of the family of maps (x, y) (x^2 + a, 0), there exists a parameter set of positive Lebesgue measure at which these maps display a non-uniformly hyperbolic attractor. This gives in particular an alternative proof of the Benedicks-Carleson Theorem."--