Non-trivial wandering domains for heterodimensional cycles

TL;DR

This paper provides conditions under which three-dimensional diffeomorphisms with heterodimensional cycles can be approximated by those with non-trivial wandering domains, using perturbations related to homoclinic tangencies and bifurcations.

Contribution

It introduces a method to approximate heterodimensional cycle diffeomorphisms by those with non-trivial wandering domains through generalized homoclinic tangencies.

Findings

Heterodimensional cycles can be approximated by diffeomorphisms with wandering domains.

Generalized homoclinic tangencies serve as organizing centers for bifurcations.

Non-trivial wandering domains are constructed via Denjoy-like methods.

Abstract

We present a sufficient condition for three-dimensional diffeomorphisms having heterodimensional cycles to be approximated arbitrarily well by diffeomorphisms with non-trivial contracting wandering domains via several perturbations. The key idea is to show that diffeomorphisms with heterodimensional cycles associated with saddle points with non-real eigenvalues can be approximated by diffeomorphisms with generalized homoclinic tangencies presented by Tatjer. The generalized homoclinic tangency is an organizing center including a Bogdanov-Takens bifurcation, by which one can obtain non-trivial contracting wandering domains together with a Denjoy-like construction.