Abstract
This article generalizes the nonexistence of wandering domains from unimodal maps to infinitely period-doubling renormalizable Hénon-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 Hénon-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 Hénon-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.
Highlights
This article studies the existence of wandering domains for Hénon-like maps
The maps are a generalization of classical Hénon maps [36] to the analytic settings and an extension of unimodal maps to higher dimensions
We show that an expansion estimate from unimodal maps can be promoted to Hénon-like maps
Summary
This article studies the existence of wandering domains for Hénon-like maps. A relevant work by Kiriki and Soma [42] found Hénon-like maps having wandering domains by using a homoclinic tangency of some saddle fixed point [41,43]. A wandering domain is a nonempty open set that is disjoint from the stable manifolds of all saddle periodic points This definition is equivalent to the classical notion of wandering intervals in the unimodal setting (see Remark 6.2). Theorem A strongly dissipative infinitely period-doubling renormalizable Hénon-like map does not have wandering domains. Hénon-like map, ω-limit sets are classified into two categories [28,48]: a saddle periodic orbit or the renormalization Cantor set that is conjugate to the dyadic adding machine From this dichotomy, a wandering domain is equivalently a nonempty open subset of the basin of the Cantor attractor.
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