As a corollary, we deduce that in generic, one-parameter 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. ![]() We also prove that the global stable sets are C^1 immersed (codimension one) submanifolds as well, provided r ge 3+ alpha with alpha close to one. 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^ for some beta >0. 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 show that in this space the bounded-type limit sets of the renormalization operator have an invariant hyperbolic structure provided r ge 2+ alpha with alpha close to one. 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.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |