|Speaker :||Professor Dyi-Shing Ou (Polish Academy of Sciences)|
|Title :||Nonexistence of Wandering Domains for Infinitely Renormalizable Henon Maps|
|Time :||2020-01-03 (Fri) 15:00 - 16:00|
|Place :||Seminar Room 722, Institute of Mathematics (NTU Campus)|
Hénon-like maps are generalizations of unimodal maps from one to two dimensions. It is known that unimodal maps do not have a wandering domain.
The main difficulty of generalizing the theorem to Hénon-like maps comes from
nonhyperbolicity . In this talk, we consider a type of Hénon-like maps, called
infinitely renormalizable maps with stationary combinatorics [3, 4]. I will explain how to resolve the problem that comes from nonhyperbolicity and prove
the theorem: an infinitely renormalizable Hénon-like map with stationary combinatorics do not have a wandering domain [1, 2]. This solves an open problem
proposed by van Strien (2010)  and Lyubich and Martens (2011) . 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.