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

TL;DR

This paper proves that in the space of $C^{r}$-diffeomorphisms on closed surfaces, non-trivial wandering domains with historic behavior are dense near homoclinic tangencies, addressing longstanding open problems in dynamical systems.

Contribution

It demonstrates the existence and density of non-trivial wandering domains with historic behavior near homoclinic tangencies in $C^{r}$-diffeomorphisms, solving key open problems.

Findings

Any Newhouse open set is in the closure of diffeomorphisms with non-trivial wandering domains.

Existence of wandering domains with historic behavior near homoclinic tangencies.

Provides a $C^{r}$-category answer to an open problem of van Strien.

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.