Zoology in the Hénon family: twin babies and Milnor's swallows
We study $C^{d,r}$-Hénon-like families $(f_{a\, b})_{a\, b}$ 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 $f_{a\, b}$ 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 $f_{a\, b}$ displays exactly two different renormalized Hénon-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) Hénon map with exactly $2$ attracting cycles (an answer to a Question by Lyubich). The proof is based on a generalization of puzzle pieces for Hénon-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.