Takens' last problem and existence of non-trivial wandering domains
Takens' last problem and existence of non-trivial wandering domains
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DOI：
10.1016/j.aim.2016.10.019
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发表时间：
2015-03
影响因子：
1.7
通讯作者：
Shin Kiriki;Teruhiko Soma
中科院分区：
文献类型：
--
作者：
Shin Kiriki;Teruhiko Soma
本文对Takens在[42]中提出的一个与动力系统的历史行为有关的公开问题给出了一个Cr（2≤ r<∞）形式的解答.为了得到答案，我们证明了在同宿切线附近存在非平凡游荡域，这是由Colli-Vargas [6，§ 2]证明的。具体地说，证明了闭曲面上Cr-同态空间中的任何纽豪斯开集都包含在具有非平凡游荡域且其前向轨道具有历史行为的C-同态集的闭包中.此外，这个结果还包含了货车Strien [39]关于Hénon族游荡域的一个公开问题在Cr范畴内的一个答案.
In this paper, we give an answer to a C r (2≤ r<∞) version of the open problem of Takens in [42] 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 [6, § 2]. Concretely speaking, it is proved that any Newhouse open set in the space of C r-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 C r category to one of the open problems of van Strien [39] which is concerned with wandering domains for Hénon family.