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Blender-producing mechanisms and a dichotomy for local dynamics for heterodimensional cycles

Dongchen Li

Abstract

Blenders are special hyperbolic sets used to produce various robust dynamical phenomena which appear fragile at first glance. We prove for $C^r$ diffeomorphisms ($r=2,\dots,\infty,ω$) that blenders naturally exist (without perturbation) near non-degenerate heterodimensional cycles of coindex-1, and the existence is determined by arithmetic properties of moduli of topological conjugacy for diffeomorphisms with heterodimensional cycles. In particular, we obtain a $C^r$-generic dichotomy for dynamics in any small neighborhood $U$ of a non-degenerate heterodimensional cycle: either there exist infinitely many blenders accumulating on the cycle, forming robust heterodimensional dynamics in most cases, or there are no orbits other than those constituting the cycle lying entirely in $U$.

Blender-producing mechanisms and a dichotomy for local dynamics for heterodimensional cycles

Abstract

Blenders are special hyperbolic sets used to produce various robust dynamical phenomena which appear fragile at first glance. We prove for diffeomorphisms () that blenders naturally exist (without perturbation) near non-degenerate heterodimensional cycles of coindex-1, and the existence is determined by arithmetic properties of moduli of topological conjugacy for diffeomorphisms with heterodimensional cycles. In particular, we obtain a -generic dichotomy for dynamics in any small neighborhood of a non-degenerate heterodimensional cycle: either there exist infinitely many blenders accumulating on the cycle, forming robust heterodimensional dynamics in most cases, or there are no orbits other than those constituting the cycle lying entirely in .
Paper Structure (27 sections, 28 theorems, 122 equations, 2 figures)

This paper contains 27 sections, 28 theorems, 122 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Blenders and moduli of topological conjugacy
  3. Simple hyperbolic dynamics
  4. Robust heterodimensional dynamics arisen from perturbations
  5. A dichotomy for local dynamics near heterodimensional cycles
  6. Plan of the proofs
  7. Results for different types of heterodimensional cycles
  8. First return maps
  9. Non-degeneracy conditions
  10. Saddle-focus case
  11. Double-focus case
  12. Blenders near saddle-focus heterodimensional cycles
  13. A criterion for the existence of blenders
  14. Existence of blenders when at least one of $\theta$ and $\dfrac{\omega}{2\pi}$ is irrational
  15. Homoclinic relations in the saddle-focus case
  16. ...and 12 more sections

Key Result

Theorem A

Let $f\in Di\! f\! f ^r(\mathcal{M})$, with $r=2,\dots,\infty,\omega$, have a non-degenerate heterodimensional cycle, and let $U$ be any small neighborhood of the cycle. In each of the above cases, the blender has index $d_1$ if it is center-stable, and index $d_2$ if it is center-unstable.

Figures (2)

  • Figure 1: LT21. A heterodimensional cycle for a three-dimensional diffeomorphism. It consists of two fixed points $L_1$ and $L_2$, a fragile heteroclinic orbit $\Gamma^0$ (blue dots) in the non-transverse intersection $W^u(L_1)\cap W^s(L_2)$, and a robust heteroclinic orbit $\Gamma^1$ (green dots) in the transverse intersection $W^s(L_1)\cap W^u(L_2)$ (red curves).
  • Figure 2: A heterodimensional cycle satisfying \ref{['word:C1']}--\ref{['word:C3']} in the straightened coordinates. Here $W^u_{loc}(O_1)$ is the vertical $y$-axis, $W^s_{loc}(O_1)$ is the horizontal $(x,z)$-plane, $W^s_{loc}(O_2)$ is the vertical $v$-axis, $W^u_{loc}(O_2)$ is the horizontal $(u,w)$-plane, $\ell^{ss}$ is a strong-stable leaf through $M^+_1$, $\ell^{uu}$ is a strong-unstable leaf through $M^-_2$. The two red curves are $\ell_1=W^s_{loc}(O_1)\cap F_{21}(W^u_{loc}(O_2))$ and $\ell_2=F^{-1}_{21}\ell_1$.

Theorems & Definitions (53)

  • Definition 1.1: Blenders
  • Definition 1.2: Heterodimensional dynamics
  • Definition 1.3: Heterodimensional cycles
  • Theorem A
  • Theorem : LT21
  • Corollary A
  • Definition 1.4: Simple hyperbolic dynamics
  • Theorem B
  • Definition 1.5: Robust heterodimensional dynamics
  • Theorem C
  • ...and 43 more