Math Seminars.
Jean-Christophe Yoccoz, College de France
Non Uniformly Hyperbolic Dynamics and Homoclinic
Bifurcations
In a jointwork with J. Palis, we consider smooth surface diffeomorphisms
exhibiting a horseshoe with an outside homoclinic tangency. When the
dimension of the horseshoe is smaller than 1, the maximal invariant set in a
neighbourhood of the union of the horseshoe and the tangency orbit is still
hyperbolic for most diffeomorphisms close to the bifurcation, according to
previous work of PALIS and TAKENS. We assume that the dimension of the
horseshoe is larger than one, but not much larger. We then show that for
most diffeomorphisms after the bifurcation, the maximal invariant set is a
"non uniformly hyperbolic horseshoe" : this object is the analogue, for
saddle-like dynamics, of Henon-like attractors.