An example of $C^1$-generically wild homoclinic classes with index deficiency

TL;DR

This paper constructs a smooth four-dimensional dynamical system with a homoclinic class that is generically wild and lacks dominated splittings, yet all periodic points have a uniform unstable index, highlighting complex dynamical behavior.

Contribution

It provides a novel example of a $C^1$-generic homoclinic class with index deficiency and no dominated splittings in four dimensions.

Findings

Homoclinic class with no dominated splittings

All periodic points have the same unstable index

The class exhibits $C^1$-generic wild behavior

Abstract

Given a closed smooth four-dimensional manifold, we construct a diffeomorphism that has a homoclinic class whose continuation locally generically satisfies the following condition: it does not admit any kind of dominated splittings whereas any periodic points belonging to it never have unstable index one.