Non-Uniformly Hyperbolic Horseshoes Arising from Bifurcations of Poincar\'e Heteroclinic Cycles

TL;DR

This paper investigates the dynamics of bifurcations in Poincaré heteroclinic cycles, revealing that most nearby systems exhibit non-uniform hyperbolicity without attractors or repellors, especially when the underlying horseshoe has Hausdorff dimension greater than one.

Contribution

It introduces the analysis of non-uniform hyperbolicity in systems with horseshoes of Hausdorff dimension exceeding one arising from heteroclinic bifurcations, extending previous understanding.

Findings

Most bifurcating systems are non-uniformly hyperbolic near the horseshoe and tangency.

Such systems lack attractors and repellors in the neighborhood.

The results apply to systems with horseshoes of Hausdorff dimension greater than one.

Abstract

The purpose of this paper is to advance the knowledge of the dynamics arising from the creation and subsequent bifurcation of Poincar\'e heteroclinic cycles. The problem is central to dynamics: it has to be addressed if, for instance, one aims at describing the typical orbit behaviour of a typical system, thus providing a global scenario for the ensemble of dynamical systems - see the Introduction and [P1, P2]. Here, we shall consider smooth, i.e. $C^\infty$, one-parameter families of dissipative, meaning non-conservative, surface diffeomorphisms. An hetereoclinic cycle may appear when the parameter evolves and an orbit of tangency, say quadratic, is created between stable and unstable manifolds (lines) of periodic orbits that belong to a basic hyperbolic set. The key novelty is to allow this basic set, a horseshoe, to have Hausdorff dimension bigger than one. In the present paper we do assume such a dimension to be beyond one, but in a limited way, as explicitly indicated in the Introduction. [A mild non-degeneracy condition on the family of maps is assumed: at the orbit of tangency the invariant lines, stable and unstable, cross each other with positive relative speed]. We then prove that most diffeomorphisms, corresponding to parameter values near the bifurcating one, are non-uniformly hyperbolic in a neighborhood of the horseshoe and the orbit of tangency; such diffeomorphisms display no attractors nor repellors in such a neighborhood. A first precise formulation of our main theorem is at the Introduction and a more encompassing version at the end of the paper. These results were announced in [PY3].