Nonexistence of Wandering Domains for Infinitely Renormalizable H\'enon Maps

TL;DR

This paper proves that infinitely renormalizable Hénon maps in the dissipative regime do not have wandering domains, extending a key result from one-dimensional unimodal maps to certain two-dimensional systems and resolving an open problem.

Contribution

It introduces new techniques to prove the nonexistence of wandering domains for Hénon-like maps, addressing a significant open problem in higher-dimensional dynamics.

Findings

Wandering domains do not exist for the class of maps studied.

The union of stable manifolds for all periodic points is dense.

The proof adapts techniques from unimodal maps to higher dimensions.

Abstract

This article extends the theorem of the absence of wandering domains from unimodal maps to infinitely period-doubling renormalizable H\'enon-like maps in the strongly dissipative (area contracting) regime. The theorem solves an open problem proposed by several authors [van Strien (2010) and Lyubich and Martens (2011)], and covers a class of maps in the nonhyperbolic higher dimensional setting. The classical proof for unimodal maps breaks down in the H\'enon settings, and two techniques, "the area argument" and "the good region and the bad region", are introduced to resolve the main difficulty. The theorem also helps to understand the topological structure of the heteroclinic web for such kind of maps: the union of the stable manifolds for all periodic points is dense.