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

Shin Kiriki, Teruhiko Soma

Published 2015-03-21Version 1

In this paper, we give an answer to a version of the open problem of Floris 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 and Vargas [Ergod. Th. & Dynam. Sys., 21 (2001), 1657-1681]. Concretely speaking, it is proved that any Newhouse open set in the $C^{r}$ topology of two-dimensional diffeomorphisms with $r \geq 2$ 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.