Zoology in the H\'enon family: twin babies and Milnor's swallows

We study $C^{d,r}$-H'enon-like families $(f_{ab}{)}_{ab}$ with two parameters $(a,b)\in {\mathbb{R}}^2$. We show the existence of an open set of parameters $(a,b)\in \mathcal{D}$, so that a renormalization chart conjugates an iterate of $f_{ab}$ to a perturbation of $(x,y)\mapsto ((x^2+c_1{)}^2+c_2,0)$. We prove that the map $(a,b)\in \mathcal{D}\mapsto (c_1,c_2)$ is a $C^d$-diffeomorphism; as first numerically conjectured by Milnor in 1992. Furthermore, we show the existence of an open set of parameters $(a,b)$ so that $f_{ab}$ displays exactly two different renormalized H'enon-like maps whose basins union attracts Lebesgue a.e. point with bounded forward orbit. A great freedom in the choice of the renormalized parameters enables us to deduce in particular the existence of a (unperturbed) H'enon map with exactly $2$ attracting cycles (an answer to a Question by Lyubich). The proof is based on a generalization of puzzle pieces for H'enon-like maps, and on a generalization of both the affine-like formalism of Palis-Yoccoz and the cross map of Shilnikov. The distortion bounds enable us to define (for the first time) $C^r$ and $C^{d,r}$-renormalizations and multi-renormalizations with bounds on all the derivatives.