Historic behavior for non-trivial wandering domains: proof of Colli-Vargas' conjecture & answer to Takens' last problem)

2016 Fiscal Year Final Research Report

• Project/Area Number: 25400112
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Tokai University
• Principal Investigator: Kiriki Shin 東海大学, 理学部, 教授 (50277232)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Keywords: ホモクリニック分岐 / ヒストリック性 / エノン写像系 / 非双曲力学系 / 非自明遊走領域 / 双曲性

Outline of Final Research Achievements

We give an answer to a smooth version of the open problem of Takens in [Nonlinearity 21 (3) (2008) T33--T36.] which is related to historic behavior of dynamical systems. To obtain the answer, we show the existence of non-trivial wandering domains near a homoclinic tangency, which is conjectured by Colli-Vargas [Ergod. Th. Dynam. Sys. 21 (2001) 1657--1681].
Concretely speaking, it is proved that any Newhouse open set in the space of smooth diffeomorphisms on a closed surface is contained in the closure of the set of diffeomorphisms which have non-trivial wandering domains whose forward orbits have historic behavior. Moreover, this result implies an answer in the smooth category to one of the open problems of van Strien [Discrete Contin. Dyn. Syst. 27 (2) (2010) 557--588] which is concerned with wandering domains for Henon family.

Free Research Field

力学系