Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Marco Martens, Liviana Palmisano

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.

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

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.

Paper Structure

This paper contains 48 sections, 53 theorems, 322 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Theorem A.
  3. Theorem B.
  4. Theorem C.
  5. Conjecture.
  6. Acknowledgements.
  7. Preliminaries
  8. Unfoldings
  9. Rescaled Families
  10. Critical Points and Critical Values
  11. Properties of unfoldings
  12. Estimates of the stable manifold at $q_2$.
  13. The $\Gamma$ curve and the $a_n$ curve.
  14. Form of the map $F^N$.
  15. Choice of $\theta$
  16. ...and 33 more sections

Key Result

Theorem 2.0.1

Given $\left(\mu, \lambda\right)\in\mathbb{R}^{2}$, there exists $k\in{\mathbb N}$ such that the following holds. Let $\mathcal{M}$ be a two dimensional ${{\mathcal{C}}^\infty}$ manifold and let $f:\mathcal{M}\to\mathcal{M}$ be a diffeomorphism with saddle point $p\in\mathcal{M}$ having unstable eig for $k=(k_1,k_2)\in{\mathbb N}^2$ and $2\leq |k|=k_1+k_2 \leq k$, then $f$ is ${{\mathcal{C}}^4}$ l

Figures (13)

  • Figure 1: A map with a strong homoclinic tangency
  • Figure 2: Unfolding
  • Figure 3: $N+\theta n+N$ iterates
  • Figure 4: $2N+n$ iterates
  • Figure 5: Position of $z^{(3)}$
  • ...and 8 more figures

Theorems & Definitions (132)

  • Theorem 2.0.1
  • Definition 2.0.3
  • Theorem 2.0.4
  • Definition 2.1.1
  • Remark 2.1.2
  • Remark 2.1.3
  • Remark 2.1.4
  • Remark 2.1.6
  • Lemma 2.1.9
  • proof
  • ...and 122 more