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On the closure of a plane ray that limits onto itself

David S. Lipham

TL;DR

This paper tackles the question of how intricate planar continua can become when they contain self-entwined rays. By developing a framework of arcs, minimality, and two-sided limit points, the author proves that the closure $\overline X$ of any self-entwined ray $X$ in the plane $\mathbb{R}^2$ contains a Cantor set of mutually disjoint subcontinua, making $\overline X$ non-Suslinian. The argument hinges on a Cantor-tree construction that yields uncountably many disjoint continua stretching from a fixed compact set to an exterior disc, under the assumption that $\overline X$ is connected im-kleinen at a dense set of points; if this assumption fails, the result follows more directly. The plane-specific result highlights a qualitative difference from higher dimensions, where analogous conclusions may fail, as evidenced by known 3D constructions and related results by Tymchatyn and Curry. Overall, the work elucidates how planar self-entwinement enforces highly non-Suslinian behavior in closures, with implications for chaotic and folding-curve dynamics in planar attractors.

Abstract

We show that the closure of any self-entwined ray in the plane must contain a Cantor set of mutually disjoint continua. This is false in dimension three.

On the closure of a plane ray that limits onto itself

TL;DR

This paper tackles the question of how intricate planar continua can become when they contain self-entwined rays. By developing a framework of arcs, minimality, and two-sided limit points, the author proves that the closure of any self-entwined ray in the plane contains a Cantor set of mutually disjoint subcontinua, making non-Suslinian. The argument hinges on a Cantor-tree construction that yields uncountably many disjoint continua stretching from a fixed compact set to an exterior disc, under the assumption that is connected im-kleinen at a dense set of points; if this assumption fails, the result follows more directly. The plane-specific result highlights a qualitative difference from higher dimensions, where analogous conclusions may fail, as evidenced by known 3D constructions and related results by Tymchatyn and Curry. Overall, the work elucidates how planar self-entwinement enforces highly non-Suslinian behavior in closures, with implications for chaotic and folding-curve dynamics in planar attractors.

Abstract

We show that the closure of any self-entwined ray in the plane must contain a Cantor set of mutually disjoint continua. This is false in dimension three.

Paper Structure

This paper contains 12 sections, 5 theorems, 5 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Terminology
  3. Motivation and main result
  4. Context and limitations
  5. Preliminaries
  6. Arcs of rays
  7. Definition of self-entwined
  8. Connectedness im-kleinen
  9. Discs and theta curves
  10. Notation
  11. Rays that limit to arcs on both sides
  12. Proof of Theorem \ref{['t1']}

Key Result

Theorem 1

If $X$ is a self-entwined ray in $\mathbb R^2$, then $\overline X$ is non-Suslinian.

Figures (5)

  • Figure 1: The Ikeda map $f(z)=1+0.9z\exp(i(0.4-\frac{6}{1+|z|^2}))\space$produces a strange attractor $K$ which contains a fixed point $p\approx 1.114- 2.285i$. The arc component of $p$ in $K$ is a self-entwined ray with initial point $p$. Iterates of $I=\{x-2.285i:0\leq x\leq 1.114\}\space$approximate initial segments of the ray since $f$ is injective on $I$, and $I\setminus \{p\}$ is contained in the basin of attraction for $K$. The arcs $f^6[I]$ and $f^7[I]$ are shown above.
  • Figure 2: A planar Plykin attractor $P$ is generated by the spherical equations of kun, followed by a stereographic projection from the point $\langle 1/\sqrt{2},0,1/\sqrt{2}\rangle$. As with the Ikeda attractor, approximations of a self-entwined ray in $P$ are obtained by iteration on an arc $I$ which ends at a fixed point of the attractor (in planar coordinates the fixed point is $p\approx \langle 1.26,-1.53\rangle$). This self-entwined ray is remarkable in that it has bounded curvature.
  • Figure 3: Illustration for Proposition \ref{['p4']}.
  • Figure 4: Construction of $D_{\langle 0\rangle}$ and $D_{\langle 1\rangle}$.
  • Figure 5: Two possible arrangements of $D_\sigma$ ($\sigma\in 2^2$) within $D_{\langle 0\rangle}$ and $D_{\langle 1\rangle}$.

Theorems & Definitions (12)

  • Theorem 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof : Idea of proof
  • Proposition 4
  • proof : Sketch of proof
  • Lemma 5
  • proof
  • Remark 1
  • ...and 2 more