# Boettcher Coordinate

Robert P. Munafo, 2023 Jul 18.

The Böttcher coordinate is the value related to the external angle (or external argument) of a point emerging from the iteration orbit dynamics of some iteration formula. It is defined along with a "Böttcher's equation" in the 1993 Carleson and Gamelin book Complex Dynamics. With regards to the Mandelbrot set specifically, it is cited in papers by Buff and Romera, as well as this wikibooks article).

The wikibooks article uses a different product "series" from the Romera paper. Romera 2013 states:

The computer programs to draw external rays of the Mandelbrot set use the Böttcher coordinate given by 4

Φ(c) = c PRODUCT_{(n=1..infinity)}
[1 + c/(f_{c}^{n-1}(c))^{2}]^{1/2n} (1a)

where c is the complex coordinate of a point outside ℳ
and f_{c}^{0}(c)=c, f_{c}^{1}(c)=c^{2}+c,
f_{c}^{2}(c)=(c^{2}+c)^{2}+c, ... are the iterates of
f_{c}(z)=z^{2}+c from the initial value z_{0}=0.

The wikibooks article, citing Wolf Jung without a specific reference, states:

To compute Boettcher coordinate w use this formula:

w = Φ_{c}(z) = z PRODUCT_{(n=1..infinity)}
[f_{c}^{n}(z)/(f_{c}^{n}(z)-c)]^{1/2n} (1b)

It looks "simple", but: square root of complex number gives two values so one have to choose one value; and more precision is needed near Julia set coordinate

In this second formula (1b), the terms of the infinite product have
f_{c}^{n}(z)-c in the denominator, where f_{c}^{n}(z) simply
refers to the iterates z_{n+1}. Therefore the first z is
z_{1} by the iterates definition, and f_{c}^{n}(z)-c is
z_{(n+1)}-c.

Since z_{(n+1)} is z_{n}^{2}+c, we can change
z_{(n+1)}-c to z_{n}^{2}. Formula (1b) becomes:

z_{1} PRODUCT_{(n=1..infinity)} [z_{n+1}/z_{n}^{2}]^{1/2n}

= z_{1} (z_{2}/z_{1}^{2})^{(1/2)} (z_{3}/z_{2}^{2})^{(1/4)}
(z_{4}/z_{3}^{2})^{(1/8)} ...

Since z_{1}=c, z_{2}=c^{2}+c, z_{3}=(c^{2}+c)^{2}+c,
and so on, it becomes:

c ((c^{2}+c)/c^{2})^{1/2}
(((c^{2}+c)^{2}+c)/(c^{2}+c)^{2})^{1/4}
((((c^{2}+c)^{2}+c)^{2}+c)/((c^{2}+c)^{2}+c)^{2})^{1/8}
... (2)

Now in each term we have something like
(bigpoly^{2}+c)/(bigpoly^{2}) so we can turn this into the sum
of two fractions (bigpoly^{2})/(bigpoly^{2}) +
c/(bigpoly^{2}), the first of which is 1 so the whole term is just
1+c/(bigpoly^{2}):

c (1 + c/c^{2})^{1/2}
(1 + c/(c^{2}+c)^{2})^{1/4}
(1 + c/((c^{2}+c)^{2}+c)^{2})^{1/8} ... (2)

Recalling again that z_{1}=c, z_{2}=c^{2}+c, etc.
we can put it back into product form as

Φ(c) = c PRODUCT_{(n=1..infinity)}[(1+c/(z_{n}^{2}))^{2n}]

with z_{n} defined as in the iterates article. Now compare this to
(1a) above, and remember that the indices z_{n} are off by 1 as compared to
the f_{c}^{n-1}(c)

Working backwards to (2) then continuing, we can work towards a series of nested square roots:

Φ(c) = c sqrt( (1+c/c^{2})
sqrt( (1+c/(c^{2}+c)^{2})
sqrt( (1+c/((c^{2}+c)^{2}+c)^{2})
... ) ) )

= c ((c^{2}+c)/c^{2})^{1/2}
(((c^{2}+c)^{2}+c)/(c^{2}+c)^{2})^{1/4}
((((c^{2}+c)^{2}+c)^{2}+c)/((c^{2}+c)^{2}+c)^{2})^{1/8}
...

= c (z_{2}/(z_{2}-c)^{1/2}
(z_{3}/(z_{3}-c)^{1/4}
(z_{4}/(z_{4}-c)^{1/8}
...

= c sqrt(z_{2}/(z_{2}-c) sqrt(z_{3}/(z_{3}-c)
sqrt(z_{4}/(z_{4}-c) ... ) ) )

We can see from these expansions that if c has a period then the
Böttcher coordinate Φ(c) is undefined because of a division by
zero, for example if z_{4}=c then (z_{4}-c) is zero.

Φ(c) is a complex number; as such it has a Magnitude Φ(c), and an Angle arg(Φ(c)). The "external ray" for a given angle a is the set of all points c for which arg(Φ(c))/π=a.

revisions: 20230718 first version (content taken from External Angle article)

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.

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