Almost every real quadratic map is either regular or stochastic

• 成果类型: Article
• 署名作者: Lyubich, M
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.2307/3597183
• 发表日期: 2002
• 页码: 1-78
• 关键词: s-unimodal maps; DYNAMICS; UNIVERSALITY; RENORMALIZATION; POLYNOMIALS; Iterations; attractors; families; RIGIDITY; bounds

摘要

In this paper we complete a program to study measurable dynamics in the real quadratic family. Our goal was to prove that almost any real quadratic map P-c : z --> x(2) + c, c is an element of [-2.1/4], has either an attracting cycle or an absolutely continuous invariant measure. The final step, completed here. is to prove that the set of infinitely renormalizable parametric values c is an element of [-2, 1/4] has zero Lebesgue measure. We derive this from a Renormalization Theorem which asserts uniform hyperbolicity of the full renormalization operator. This theorem gives the most general real version of the Feigenbaum-Coullet-Tresser universality. simultanuously for all combinatorial types.