Serie : Astérisque. Vol 410
Paru le 15/06/2019 | Broché VII-177 pages
Professionnels
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→x2 + 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 C2-perturbation of the family of maps (x,y) → (x2 + 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.
