[Paper Review] Non-Uniformly Hyperbolic Horseshoes Arising from Bifurcations of Poincaré Heteroclinic Cycles

Jacob Palis, Jean-Christophe Yoccoz|ArXiv.org|Apr 7, 2006

Mathematical Dynamics and Fractals4 citations

TL;DR

This paper investigates non-uniformly hyperbolic horseshoes emerging from bifurcations of Poincaré heteroclinic cycles in smooth, one-parameter families of dissipative surface diffeomorphisms. By allowing the horseshoe to have Hausdorff dimension greater than one and assuming a mild non-degeneracy condition on tangency, the authors prove that most parameter values near the bifurcation yield non-uniformly hyperbolic dynamics with no attractors or repellors in the neighborhood of the horseshoe and tangency orbit.

ABSTRACT

The purpose of this paper is to advance the knowledge of the dynamics arising from the creation and subsequent bifurcation of Poincaré 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].

Motivation & Objective

• To understand the dynamics arising from the creation and bifurcation of Poincaré heteroclinic cycles, a central problem in dynamical systems.
• To analyze the emergence of non-uniformly hyperbolic behavior in systems where the basic hyperbolic set (a horseshoe) has Hausdorff dimension greater than one.
• To establish conditions under which such systems avoid the formation of attractors or repellors near the horseshoe and tangency orbit.
• To provide a global scenario for typical dynamical systems by characterizing the behavior of a generic one-parameter family near a bifurcation point.

Proposed method

• The analysis uses a Markov partition and defines a folding map G to model the dynamics near the tangency and horseshoe.
• Affine-like maps are employed to represent the dynamics, with cone conditions and distortion control to ensure hyperbolicity.
• A transversality relation is defined via a critical distance δ(Q,P′) ≥ 2 max(|Q|^{1−η}, |P′|^{1−η}) to ensure robust non-degeneracy.
• The construction of parameter intervals relies on a selection process and strong regularity conditions to control the geometry of rectangles and their iterates.
• The method introduces a hereditary and concave transversality relation, replacing the initial non-concave set with a larger, well-behaved class of rectangles.
• A spectral analysis of the transfer operator and the construction of a Gibbs measure are used to estimate the Hausdorff dimension of the invariant set.

Experimental results

Research questions

• RQ1 Under what conditions does a heteroclinic cycle bifurcation lead to non-uniformly hyperbolic dynamics in surface diffeomorphisms?
• RQ2 How does the Hausdorff dimension of the horseshoe influence the emergence of non-uniform hyperbolicity after bifurcation?
• RQ3 What role does the relative speed of stable and unstable manifolds at tangency play in determining the dynamical behavior near the bifurcation?
• RQ4 Can the absence of attractors or repellors be rigorously established in the neighborhood of the horseshoe and tangency orbit for generic parameter values?
• RQ5 What geometric and dynamical structures (e.g., parabolic cores, critical rectangles) emerge in the invariant set under the given bifurcation?

Key findings

• For most parameter values near the bifurcation, the diffeomorphisms are non-uniformly hyperbolic in a neighborhood of the horseshoe and the orbit of tangency.
• The dynamics exhibit no attractors or repellors in this neighborhood, implying the absence of periodic sinks or sources.
• The Hausdorff dimension of the invariant set is estimated via a transfer operator and Gibbs measure, with the transverse Hausdorff dimension of the invariant set being positive and finite.
• The construction of the parameter space relies on a strong regularity condition, ensuring that the number of critical rectangles and their geometric distortion remain under control.
• The transversality relation is refined into a hereditary and concave form, enabling inductive control over the geometry of iterated rectangles.
• The exceptional set E(ω*) has Hausdorff dimension strictly less than the full dimension of the invariant set, indicating a non-uniform structure.