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.