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

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.