2018/01/17 by Pierre Berger, Berger, Pierre · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1801.05628
openalex publication_date 2018/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study Cd,r-Hénon-like families (fa b)a b with two parameters (a,b)∈ \mathbb R2. We show the existence of an open set of parameters (a,b)∈ \mathcal D, so that a renormalization chart conjugates an iterate of fa b to a perturbation of (x,y)↦ ((x2+c1)2+c2,0). We prove that the map (a,b)∈ \mathcal D↦ (c1,c2) is a Cd-diffeomorphism; as first numerically conjectured by Milnor in 1992. Furthermore, we show the existence of an open set of parameters (a,b) so that fa 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) Cr and Cd,r-renormalizations and multi-renormalizations with bounds on all the derivatives.