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)
Abstract: 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 [6]. 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) [5] and Lyubich and Martens (2011) [6]. 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.

[1] D. Ou, Nonexistence of wandering domains for infinitely renormalizable Hénon maps, Stony Brook University PhD Thesis (2018)
[2] D. Ou, Nonexistence of wandering domains for strongly dissipative infinitely renormalizable Hénon maps at the boundary of chaos, Invent. math. (2019), doi:10.1007/s00222-019-00902-4
[3] A. de Carvalho, M. Lyubich, and M. Martens, Renormalization in the Hénon family, I: Universality but non-rigidity, J. Stat. Phys. 121 (2005), 611–669.
[4] P. Hazard, Hénon-like maps with arbitrary stationary combinatorics, Ergod. Theor. Dyn. Syst. 31 (2011), 1391–1443.
[5] S. van Strien, One-dimensional dynamics in the new millennium, Discrete Cont. Dyn. S. 27 (2010), 557–588.
[6] M. Lyubich, M. Martens, Renormalization in the Hénon family, II: the heteroclinic web, Invent. Math. 186 (2011), 115–189.