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

In this paper, we give an answer to a Cr (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 Cr-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 Cr category to one of the open problems of van Strien [39] which is concerned with wandering domains for Hénon family.