Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms

TL;DR

This paper establishes a dichotomy for diffeomorphisms on compact manifolds, showing they can be approximated by either systems with homoclinic bifurcations or essentially hyperbolic systems with predictable attractors.

Contribution

It proves a new dichotomy result that classifies diffeomorphisms into those with homoclinic bifurcations or essentially hyperbolic behavior, advancing understanding of dynamical systems.

Findings

Any diffeomorphism can be approximated by one with a homoclinic bifurcation.

Alternatively, it can be approximated by an essentially hyperbolic diffeomorphism.

The result provides a clear dichotomy in the behavior of diffeomorphisms.

Abstract

We prove that any diffeomorphism of a compact manifold can be approximated in topology C1 by another diffeomorphism exhibiting a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by one which is essentially hyperbolic (it has a finite number of transitive hyperbolic attractors with open and dense basin of attraction).