Wild Sets with Collet-Eckmann Points and Infinitely Many Sinks: Stability and Coexistence

TL;DR

This paper explores the coexistence of wild non-hyperbolic Cantor sets with Collet-Eckmann points and infinitely many sinks in two-dimensional dynamical systems, revealing complex stability phenomena and introducing a new renormalization approach.

Contribution

It demonstrates the coexistence of wild hyperbolic and non-hyperbolic dynamics on invariant sets and introduces a generalized renormalization scheme for such systems.

Findings

Wild Cantor sets contain Collet-Eckmann points with dense orbits.

Existence of infinitely many sinks accumulating on wild Cantor sets.

Persistence of complex dynamical structures along codimension one manifolds.

Abstract

In two-dimensional unfoldings of homoclinic tangencies, the parameter space contains codimension one laminations whose leaves consist of maps with invariant non-hyperbolic Cantor sets. These Cantor sets are wild both in the sense of Hofbauer-Keller and in the sense of Newhouse, and they contain Collet-Eckmann points with dense orbits. Hence, wildness and non-uniform chaotic hyperbolicity can coexist on a single invariant set, while persisting along codimension one manifolds. In addition, each leaf of the lamination contains a map with infinitely many sinks accumulating on the Cantor set containing the Collet-Eckmann point. This surprising symbiosis of contraction and expansion may not, in fact, be pathological. Along the way, we introduce a generalized renormalization scheme for two-dimensional systems.