Historic wandering domains near cycles

TL;DR

This paper demonstrates the existence of non-trivial historic wandering domains near cycles in certain diffeomorphisms, addressing Takens' last problem in the $C^1$ topology and higher dimensions.

Contribution

It introduces a method to obtain historic wandering domains near cycles in $C^r$-Newhouse domains, advancing understanding of complex dynamics in higher dimensions.

Findings

Existence of historic wandering domains near cycles in $C^r$-Newhouse domains.

First contribution to Takens' last problem in $C^1$ topology for $d>2$.

Construction of these domains near diffeomorphisms with heterodimensional or non-transverse equidimensional cycles.

Abstract

We explain how to obtain non-trivial historic contractive wandering domains for a dense set of diffeomorphisms in certain type of $C^r$-Newhouse domains of homoclinic tangencies in dimension $d\geq 3$ and $r\geq 1$. In particular, this gives for the first time a contribution to Takens' last problem in the $C^1$ topology and in dimension $d>2$. We show how these Newhouse domains can be obtained arbitrarily close to diffeomorphisms exhibiting heterodimensional cycles (in dimension $d=3$) or non-transverse equidimensional cycles (in any dimension $d\ge 3$) associated with periodic points with non-real complex leading multipliers.