Steps towards a classification of $C^r$-generic dynamics close to homoclinic points

We present here the first part of a program for a classification of the generic dynamics close to homoclinic and heteroclinic points, in the $C^r$ topologies, $r\geq 1$. This paper only contains announcements and a few sketches of proofs; a forthcoming series of papers will present the proofs in details. The two prototypical examples of non-hyperbolic dynamics are homoclinic tangencies and heterodimensional cycles. Palis conjectured that they actually characterize densely non-hyperbolic dynamics. It is therefore important to understand what happens close to those bifurcations. We generalize classical results of Newhouse, Palis and Viana, for both tangencies and cycles: close to a homoclinic tangency or to a heterodimensional cycle there is abundance of diffeomorphisms exhibiting infinitely many sinks or sources if and only if the dynamics is not volume-hyperbolic. This proves in particular a conjecture of Turaev for homoclinic tangencies. An important result of Bonatti, Diaz, Pujals states that if a homoclinic class is $C^1$-robustly without dominated splitting, then nearby diffeomorphisms exhibit $C^1$-generically infinitely many sinks or sources. We show that this holds in higher regularities, under the further assumption that non-dominations are obtained through so-called "mechanisms". This includes all the examples of robustly non-dominated homoclinic classes one can build with the tools known up to now. We actually have a $C^r$-equivalent of a recent $C^1$-result of Bochi and Bonatti: we describe precisely the Lyapunov exponents along periodic points that may appear close to a homoclinic tangency or to a homoclinic class. The results of Newhouse, Palis and Viana were proven for the $C^r$ topologies, $r\geq 2$. Our results hold also in the $C^{1+\alpha}$ topologies.