Non-periodic bifurcation for surface diffeomorphisms

TL;DR

This paper investigates the boundary of hyperbolic surface diffeomorphisms, showing that a positive probability subset consists of Kupka-Smale diffeomorphisms with specific tangency properties, revealing non-periodic bifurcations.

Contribution

It demonstrates that a positive measure subset of the boundary of hyperbolic surface diffeomorphisms contains Kupka-Smale maps with non-periodic tangencies, advancing understanding of bifurcations.

Findings

Boundary of hyperbolic diffeomorphisms contains Kupka-Smale maps.

Non-hyperbolicity caused by tangencies not linked to periodic points.

Constructed diffeomorphisms lie on the boundary of a connected component.

Abstract

We prove that a "positive probability" subset of the boundary of the set of hyperbolic (Axiom A) surface diffeomorphisms with no cycles $\mathcal{H}$ is constituted by Kupka-Smale diffeomorphisms: all periodic points are hyperbolic and their invariant manifolds intersect transversally. Lack of hyperbolicity arises from the presence of a tangency between a stable manifold and an unstable manifold, one of which is not associated to a periodic point. All these diffeomorphisms that we construct lie on the boundary of the same connected component of $\mathcal{H}$.