Global hyperbolicity of renormalization for C^runimodal mappings

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 C r unimodal maps with quadratic critical point. We show that in this space the boundedtype 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 C 1 codimension one, Banach submanifolds of the ambient space, and whose holonomy is C 1+β for some β > 0. We also prove that the global stable sets are C 1 immersed (codimension one) submanifolds as well, provided r ≥ 3 + α with α close to one. As a corollary, we deduce that in generic, oneparameter families of C r 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.

Hausdorff dimension bounds for smoothness of holonomies for codimension 1 hyperbolic dynamics

“…For simplicity of our exposition, we consider Markov partitions that satisfy the disjointness property. This result is also used in [5][6][7]18,[21][22][23]25].…”

Complex Bounds for Real Maps

“…that these bounds are beau, see p. 5. This means that there exists a compact class C of maps, so that after renormalizing a map f , possibly a large number of times, its renormalization is in C. Subsequent renormalization results, including exponential convergence of renormalisation, were obtained by [M1,M2,Ly3,Ly5,dFdMP]. The most recent proof in [AL] of the convergence of renormalization shows that, in fact, the property of complex bounds with beau estimates is essentially the only ingredient that is required.…”

On the Invariant Cantor Sets of Period Doubling Type of Infinitely Renormalizable Area-Preserving Maps

“…It has been particularly successful in one-dimensional dynamics, explaining universality for unimodal maps, critical circle maps and maps with a Siegel disk (see e.g. [1,4,12,15,18,19,21,22] and references therein).…”