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

By Shin Kiriki, Teruhiko Soma. 2015

78 citations recorded by the source corpus.

Abstract 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 Henon family.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/4f02e49ea8f8e4409ae1ec75788af2db56b4b2ee