Uniformization

Let ${\displaystyle \Psi _{c}\,}$ be the conformal isomorphism from the complement (exterior) of the closed unit disk ${\displaystyle {\overline {\mathbb {D} }}}$ to the complement of the filled Julia set ${\displaystyle \ K_{c}}$.

${\displaystyle \Psi _{c}:{\hat {\mathbb {C} }}\setminus {\overline {\mathbb {D} }}\to {\hat {\mathbb {C} }}\setminus K_{c}}$

where ${\displaystyle {\hat {\mathbb {C} }}}$ denotes the extended complex plane. Let ${\displaystyle \Phi _{c}=\Psi _{c}^{-1}\,}$ denote the Boettcher map.^[14] ${\displaystyle \Phi _{c}\,}$ is a uniformizing map of the basin of attraction of infinity, because it conjugates ${\displaystyle f_{c}}$ on the complement of the filled Julia set ${\displaystyle K_{c}}$ to ${\displaystyle f_{0}(z)=z^{2}}$ on the complement of the unit disk:

${\displaystyle {\begin{aligned}\Phi _{c}:{\hat {\mathbb {C} }}\setminus K_{c}&\to {\hat {\mathbb {C} }}\setminus {\overline {\mathbb {D} }}\\z&\mapsto \lim _{n\to \infty }(f_{c}^{n}(z))^{2^{-n}}\end{aligned}}}$

and

${\displaystyle \Phi _{c}\circ f_{c}\circ \Phi _{c}^{-1}=f_{0}}$

A value ${\displaystyle w=\Phi _{c}(z)}$ is called the Boettcher coordinate for a point ${\displaystyle z\in {\hat {\mathbb {C} }}\setminus K_{c}}$.

Formal definition of dynamic ray

Polar coordinate system and ${\displaystyle \psi _{c}}$ for ${\displaystyle c=-2}$

The external ray of angle ${\displaystyle \theta \,}$ noted as ${\displaystyle {\mathcal {R}}_{\theta }^{K}}$is:

• the image under ${\displaystyle \Psi _{c}\,}$ of straight lines ${\displaystyle {\mathcal {R}}_{\theta }=\{\left(r\cdot e^{2\pi i\theta }\right):\ r>1\}}$

${\displaystyle {\mathcal {R}}_{\theta }^{K}=\Psi _{c}({\mathcal {R}}_{\theta })}$

• set of points of exterior of filled-in Julia set with the same external angle ${\displaystyle \theta }$

${\displaystyle {\mathcal {R}}_{\theta }^{K}=\{z\in {\hat {\mathbb {C} }}\setminus K_{c}:\arg(\Phi _{c}(z))=\theta \}}$

Uniformization

Boundary of Mandelbrot set as an image of unit circle under ${\displaystyle \Psi _{M}\,}$

Uniformization of complement (exterior) of Mandelbrot set

Let ${\displaystyle \Psi _{M}\,}$ be the mapping from the complement (exterior) of the closed unit disk ${\displaystyle {\overline {\mathbb {D} }}}$ to the complement of the Mandelbrot set ${\displaystyle \ M}$.^[17]

${\displaystyle \Psi _{M}:\mathbb {\hat {C}} \setminus {\overline {\mathbb {D} }}\to \mathbb {\hat {C}} \setminus M}$

and Boettcher map (function) ${\displaystyle \Phi _{M}\,}$, which is uniformizing map^[18] of complement of Mandelbrot set, because it conjugates complement of the Mandelbrot set ${\displaystyle \ M}$ and the complement (exterior) of the closed unit disk

${\displaystyle \Phi _{M}:\mathbb {\hat {C}} \setminus M\to \mathbb {\hat {C}} \setminus {\overline {\mathbb {D} }}}$

it can be normalized so that :

${\displaystyle {\frac {\Phi _{M}(c)}{c}}\to 1\ as\ c\to \infty \,}$^[19]

where :

${\displaystyle \mathbb {\hat {C}} }$ denotes the extended complex plane

Jungreis function ${\displaystyle \Psi _{M}\,}$ is the inverse of uniformizing map :

${\displaystyle \Psi _{M}=\Phi _{M}^{-1}\,}$

In the case of complex quadratic polynomial one can compute this map using Laurent series about infinity^[20][21]

${\displaystyle c=\Psi _{M}(w)=w+\sum _{m=0}^{\infty }b_{m}w^{-m}=w-{\frac {1}{2}}+{\frac {1}{8w}}-{\frac {1}{4w^{2}}}+{\frac {15}{128w^{3}}}+...\,}$

where

${\displaystyle c\in \mathbb {\hat {C}} \setminus M}$

${\displaystyle w\in \mathbb {\hat {C}} \setminus {\overline {\mathbb {D} }}}$

Formal definition of parameter ray

The external ray of angle ${\displaystyle \theta \,}$ is:

• the image under ${\displaystyle \Psi _{c}\,}$ of straight lines ${\displaystyle {\mathcal {R}}_{\theta }=\{\left(r*e^{2\pi i\theta }\right):\ r>1\}}$

${\displaystyle {\mathcal {R}}_{\theta }^{M}=\Psi _{M}({\mathcal {R}}_{\theta })}$

• set of points of exterior of Mandelbrot set with the same external angle ${\displaystyle \theta }$^[22]

${\displaystyle {\mathcal {R}}_{\theta }^{M}=\{c\in \mathbb {\hat {C}} \setminus M:\arg(\Phi _{M}(c))=\theta \}}$

Definition of the Boettcher map

Douady and Hubbard define:

${\displaystyle \Phi _{M}(c)\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \Phi _{c}(z=c)\,}$

so external angle of point ${\displaystyle c\,}$ of parameter plane is equal to external angle of point ${\displaystyle z=c\,}$ of dynamical plane

Computation of external argument

• argument of Böttcher coordinate as an external argument^[24]
  • ${\displaystyle \arg _{M}(c)=\arg(\Phi _{M}(c))}$
  • ${\displaystyle \arg _{c}(z)=\arg(\Phi _{c}(z))}$
• kneading sequence as a binary expansion of external argument^[25][26][27]