Simultaneous Continuation of Infinitely Many Sinks Near a Quadratic Homoclinic Tangency

TL;DR

This paper demonstrates that for certain surface diffeomorphisms with infinitely many sinks near a quadratic homoclinic tangency, specific perturbations can preserve infinitely many sinks simultaneously, unlike generic one-parameter unfoldings.

Contribution

It shows how to perturb surface diffeomorphisms to maintain infinitely many sinks, revealing differences between infinite-dimensional and one-parameter perturbations.

Findings

Infinite-dimensional perturbations preserve infinitely many sinks.

One-parameter unfoldings typically preserve only finitely many sinks.

The results highlight the complexity of sink persistence near homoclinic tangencies.

Abstract

We prove that the $C^3$ diffeomorphisms on surfaces, exhibiting infinitely many sinksnear the generic unfolding of a quadratic homoclinic tangency of a dissipative saddle, can be perturbed along an infinite dimensional manifold of $C^3$ diffeomorphisms such that infinitely many sinks persist simultaneously. On the other hand, if they are perturbed along one-parameter families that unfold generically the quadratic tangencies, then at most a finite number of those sinks have continuation.