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The Parabolic Mandelbrot Set

Carsten Lunde Petersen, Pascale Roesch

Abstract

We solve the longstanding conjecture by Milnor (1993) concerning the connectedness locus $M_1$ of the family of quadratic rational maps tangent to the identity at $\infty$. We prove that this locus in homeomorphic to the Mandelbrot set $M$ and that the homeomorphism is unique, provided it identifies maps that are "hybridly" conjugate on their filled-in Julia set. Moreover this homeomorphism from $M$ to $M_1$ is nowhere Hölder on the boundary and so can not have even locally a quasi-conformal extension to complements.

The Parabolic Mandelbrot Set

Abstract

We solve the longstanding conjecture by Milnor (1993) concerning the connectedness locus of the family of quadratic rational maps tangent to the identity at . We prove that this locus in homeomorphic to the Mandelbrot set and that the homeomorphism is unique, provided it identifies maps that are "hybridly" conjugate on their filled-in Julia set. Moreover this homeomorphism from to is nowhere Hölder on the boundary and so can not have even locally a quasi-conformal extension to complements.

Paper Structure

This paper contains 33 sections, 68 theorems, 75 equations, 26 figures.

Table of Contents

  1. Introduction
  2. Background and state of the art
  3. Basic notations and description of parameter spaces.
  4. Basic notions for quadratic polynomials
  5. Wakes
  6. Basic notation for maps in $Per_1(1)$.
  7. Parametrization of $\hbox{$\mathbb C$}\setminus {\bf M_1}$
  8. Parabolic Rays
  9. Dynamical parabolic rays
  10. Parabolic ray for the Blaschke product
  11. Parabolic rays for the rational map $g_B$
  12. Limbs of ${\bf M_1}$
  13. Landing of parameter rays
  14. Wakes
  15. Parabolic Puzzles and Parabolic Para-puzzles
  16. ...and 18 more sections

Key Result

Theorem 1

There exists a dynamical holomorphic motion $\Phi : \hbox{$\mathbb D$}\times{\bf M}_0 \longrightarrow \hbox{$\mathbb C$}$ of ${\bf M}_0$ over $\hbox{$\mathbb D$}$ with base point $\mu_0 = 0$ such that $\Phi^\mu({\bf M}_0) = {\bf M}_\mu$ for all $\mu\in\hbox{$\mathbb D$}$.

Figures (26)

  • Figure 1: The parabolic connectedness locus ${\bf M_1}$ (left) and the ${\bf M}$ (right)
  • Figure 2: The sets ${\bf M}_\mu$ for some parameters $\mu \in [0,1]$.
  • Figure 3: A parabolic Julia set (left) and the corresponding one in ${\bf M}$ (right)
  • Figure 4: In black, ${\bf M_1}$, viewed in the coordinate $\sigma^1$, the product of the two remaining fixed point multipliers. Maps in $Per_1(1)$ have a degenerate fixed point. Hence $\sigma^1([g])$ is also the multiplier of the unique third fixed point $\alpha_g$ of $g$. In particular the big black disk is the unit disk, it corresponds to $\alpha_g$ being attracting.
  • Figure 5: Mandelbrot set -- the central cardioid with limbs attached
  • ...and 21 more figures

Theorems & Definitions (132)

  • Conjecture : Milnor, 1983
  • Theorem : Lyubich, Uhre, Bassanelli-Berteloot
  • Theorem : Haissinsky
  • Theorem \oldthetheorem: Douady-Hubbard
  • Theorem \oldthetheorem: Douady
  • Theorem \oldthetheorem: Douady-Hubbard
  • Theorem \oldthetheorem: Douady-Hubbard
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • ...and 122 more