Partial hyperbolicity far from homoclinic bifurcations

TL;DR

This paper demonstrates that any diffeomorphism can be approximated by either one with a homoclinic bifurcation or one that is partially hyperbolic with low-dimensional center bundles, advancing understanding of dynamical systems.

Contribution

It establishes a new approximation result connecting homoclinic bifurcations and partial hyperbolicity in dynamical systems.

Findings

Any diffeomorphism can be C^1-approximated by a partially hyperbolic one.

Introduces systematic study of central models in dynamical systems.

Provides conditions for approximating systems with homoclinic bifurcations.

Abstract

We prove that any diffeomorphism of a compact manifold can be C^1-approximated by a diffeomorphism which exhibits a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by a diffeomorphism which is partially hyperbolic (its chain-recurrent set splits into partially hyperbolic pieces whose centre bundles have dimensions less or equal to two). We also study in a more systematic way the central models introduced in arXiv:math/0605387.