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arXiv · 1503.06258

Takens' last problem and existence of non-trivial wandering domains

Shin Kiriki·Teruhiko Soma
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Abstract

In this paper, we give an answer to a $C^{r}$ $(2\leq r <\infty)$ version of the open problem of Takens in [Nonlinearity, 21 (2008), no.3, 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 $C^{r}$-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 [Discrete Conti. Dynam. Sys., 27 (2010), no.2, 557-588] which is concerned with wandering domains for H\'enon family.

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Shin Kiriki, Teruhiko Soma. 2015-03-21. Takens' last problem and existence of non-trivial wandering domains. https://doi.org/10.1016/j.aim.2016.10.019

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