Nonexistence of wandering domains for strongly dissipative infinitely renormalizable Henon maps at the boundary of chaos

• 成果类型: Article
• 署名作者: Ou, Dyi-Shing
• 署名单位: State University of New York (SUNY) System; Stony Brook University; Polish Academy of Sciences; Institute of Mathematics of the Polish Academy of Sciences
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-019-00902-4
• 发表日期: 2020
• 页码: 219-280
• 关键词: dimensional dynamical-systems; topological attractors; diffeomorphisms; fatou; mappings; THEOREM; HYPERBOLICITY; EXISTENCE; intervals; FAMILY

摘要

This article generalizes the nonexistence of wandering domains from unimodal maps to infinitely period-doubling renormalizable Henon-like maps in the strongly dissipative (area contracting) regime. This solves an open problem proposed by van Strien (Discrete Contin Dyn Syst 27(2):557-588, 2010) and Lyubich and Martens (Invent Math 186(1):115-189, 2011). We partition the phase space of a Henon-like map into two regions: the good region and the bad region. The good region is where the method of proof for unimodal maps applies to Henon-like maps, while the bad region is where serious difficulties occur. These difficulties are resolved by the Two-Row Lemma, an inequality that relates the contraction of areas to the contraction of bad regions. After analyzing the competition of the two types of contraction, we show that the case of bad regions happens at most finitely many times and complete the proof. As an application, the theorem enriches our understanding of the topological structure of the heteroclinic web: the union of the stable manifolds of periodic orbits forms a dense set in the domain.