Abstract
In this paper we extend M. Lyubich’s recent results on the global hyperbolicity of renormalization of quadratic-like germs to the space of Cr unimodal maps with quadratic critical point. We show that in this space the bounded-type limit sets of the renormalization operator have an invariant hyperbolic structure provided r≥2+α with α close to one. As an intermediate step between Lyubich’s results and ours, we prove that the renormalization operator is hyperbolic in a Banach space of real analytic maps. We construct the local stable manifolds and prove that they form a continuous lamination whose leaves are C1 codimension one, Banach submanifolds of the ambient space, and whose holonomy is C1+β for some β>0. We also prove that the global stable sets are C1 immersed (codimension one) submanifolds as well, provided r≥3+α with α close to one. As a corollary, we deduce that in generic, one-parameter families of Cr unimodal maps, the set of parameters corresponding to infinitely renormalizable maps of bounded combinatorial type is a Cantor set with Hausdorff dimension less than one.

KEYWORDS

SHARE & LIKE

COMMENTS

SIMILAR ARTICLES

Abstract For any nondegenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a correspondin

Read MoreAbstract For a large class of nonlinear Schrödinger equations with nonzero conditions at infinity and for any speed c less than the sound velocity, we

Read MoreAbstract Let L2,p(ℝ2) be the Sobolev space of real-valued functions on the plane whose Hessian belongs to Lp. For any finite subset E⊂ℝ2 and p>2, let

Read MoreAbstract We prove that for any group G in a fairly large class of generalized wreath product groups, the associated von Neumann algebra LG completely

Read MoreAbstract We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We

Read MoreAbstract This paper has two main results. Firstly, we complete the parametrisation of all p-blocks of finite quasi-simple groups by finding the so-cal

Read MoreAbstract We derive sharp Moser-Trudinger inequalities on the CR sphere. The first type is in the Adams form, for powers of the sublaplacian and for ge

Read MoreAbstract We prove that isoparametric hypersurfaces with (g,m)=(6,2) are homogeneous, which answers Dorfmeister-Neher’s conjecture affirmatively and so

Read MoreAbstract We prove the periodicity conjecture for pairs of Dynkin diagrams using Fomin-Zelevinsky’s cluster algebras and their (additive) categorificat

Read MoreAbstract If F(x,y)∈ℤ[x,y] is an irreducible binary form of degree k≥3, then a theorem of Darmon and Granville implies that the generalized superellipt

Read More