Feigenbaum-Coullet-Tresser universality and Milnor's hairiness conjecture

• 成果类型: Article
• 署名作者: Lyubich, M
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.2307/120968
• 发表日期: 1999
• 页码: 319-420
• 关键词: non-linear transformations; quadratic polynomials; holomorphic motions; metric properties; DYNAMICS; MAPS; PROOF; RENORMALIZATION; ENDOMORPHISMS; bifurcations

摘要

We prove the Feigenbaum-Coullet-Tresser conjecture on the hyperbolicity of the renormalization transformation of bounded type. This gives the first computer-free proof of the original Feigenbaum observation of the universal parameter scaling laws. We use the Hyperbolicity Theorem to prove Milnor's conjectures on self-similarity and hairiness of the Mandelbrot set near the corresponding parameter values. We also conclude that the set of real infinitely renormalizable quadratics of type bounded by some N > 1 has Hausdorff dimension strictly between 0 and 1. In the course of getting these results we supply the space of quadratic-like germs with a complex analytic structure and demonstrate that the hybrid classes form a complex codimension-one foliation of the connectedness locus.