Decay of geometry for unimodal maps: negative Schwarzian case

• 成果类型: Article
• 署名作者: Graczyk, J; Sands, D; Swiatek, G
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2005.161.613
• 发表日期: 2005
• 页码: 613-677
• 关键词: induced expansion; metric properties; sets

摘要

We show that decay of geometry holds for unimodal maps of the interval which have negative Schwarzian derivative, sufficient finite smoothness, and a nondegenerate critical point. The proof is based on pseudo-analytic extensions of order at least 2. They allow us to modify Sullivan's principle that rescaled high iterates of one-dimensional maps tend to analytic limits in such a way that no passage to a limit is actually needed, but the maps are shown to approach the analytic class in a well defined sense. As a technical improvement, this method yields a uniform estimate in the case of renormalizable maps.