2015/03/31 by Shin Kiriki, Teruhiko Soma
Mathematics · #math.DS #msc:37C29 #msc:37D20 #msc:37D25 #msc:37G25
paper · pdf · doi:10.1016/j.aim.2016.10.019
published as Advances in Mathematics, 306 (2017) pp. 524-588
arxiv created 2016/11/04 · arxiv updated 2016/11/07
In this paper, we give an answer to a Cr (2≤ r <∞) 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 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 [Discrete Conti. Dynam. Sys., 27 (2010), no.2, 557-588] which is concerned with wandering domains for Hénon family.