A few weeks ago (from July 5 to July 11) I attented the interesting conference Dynamische Systeme held at the MFO (Oberwolfach). The talks concerned several topics in Dynamical Systems and Symplectic Geometry, so that it was a nice opportunity to learn some useful techniques and theorems from (directly or indirectly related) diverse subareas.
Also, the organizers (Jean-Christophe Yoccoz, Hakan Eliasson and Eduard Zehnder) gave an opportunity to speak about a work in progress (joint with C. Gugu Moreira and E. Pujals) towards the so-called Smale conjecture. The title of the talk was “ density of hyperbolicity for Benedicks-Carleson toy models” and the goal of the talk was the proof of Smale’s conjecture in a “toy” version of the Henon dynamics introduced by M. Benedicks and L. Carleson in their seminal paper “The dynamics of Henon map“.
More precisely, Smale’s conjecture claims that hyperbolic (Axiom A) diffeomorphisms of compact surfaces are dense. Firstly, let me make some comments about this conjecture: from the works of M. Shub and R. Mane we know that Smale’s conjecture is false in higher dimensions (i.e., among diffeomorphisms of compact manifolds with dimension ). Also, S. Newhouse showed that Smale’s conjecture is also false when we replace by , : in fact, he used clever arguments involving dynamically defined Cantor sets in order to prevent hyperbolicity in a robust way via the so-called homoclinic tangencies (this is the known as Newhouse phenomena). However, after a recent work of Gugu (Moreira), we know that Newhouse’s argument (using the existence of stable intersections of Cantor sets) doesn’t work in the topology. In particular, this gives some hope towards the validity of Smale’s conjecture. For a more detailed exposition of these topics, one can see my previous posts (part I and part II) on Gugu’s theorem.
In this direction, Gugu (Moreira), Enrique (Pujals) and I decided to test some ideas around Smale’s conjecture in some toy models. Of course, we wanted some nice class of examples such that: the class is rich enough so that the Newhouse phenomena occurs in the world, but the class is simple enough so that one can apply Gugu’s results (and some geometrical arguments) to prove Smale’s conjecture in the world. Inspired by the Benedicks-Carleson work, we consider their toy models for the Henon dynamics. This class is quite promising because the geometry is simple (it is a skew-product of the quadratic family and some contracting maps on a fan of lines), although it is rich enough to exhibit the Newhouse phenomena (by the works of R. Ures).
During the talk, I discussed our ideas (related to the notion of dynamical critical points of F. Rodriguez-Hertz and E. Pujals) and how they eventually lead to a proof of Smale conjecture in this particular class.
For more details and references, please see the report of my talk appearing in this preliminary version of the Oberwolfach Report N. 32 containing the reports of all lectures of the conference (these reports are going to appear in the Oberwolfach Reports in a near future).
Finally, you can see some photos of this workshop here.