The iterated quadratic is

z_{n+1} = z_{n}^{2}+c

The renormalized iteration count is

m(R) = n+1 - log(log |z_{n}|) / log 2

where we have picked an escape radius R, and defined n to be the
smallest n such that R < |z_{n}| . Then, as shown on the
Escape Page, m(R) is
'almost' independent of R and n, in the sense that m approaches a
well-defined limit as R->oo, and, in fact, converges very rapidly.
It is also 'almost' continuous and smooth, in the sense that
there are only a measure-zero of discontinuities for finite R, and that
all discontinuities are bounded, and get vanishingly small in the limit
of R->oo. That is, in the limit, m becomes continuous and
smooth, and, as a practical, computational matter, is plenty smooth
enough for small R (100 to 10000). Below, when we write m, we sometimes
mean the limit m(R->oo) and sometimes m(R) for finite R, and most
of the statements apply to either.

The Douady-Hubbard potential is just

f = e^{-m log2} = 2^{-m}

Let us now show that f is harmonic, both in the limit R->oo and for finite R, and that it has the right asymptotic behavior. The limitations of html make this presentation hard. Please bear with us as we define some basic notation.

Lets define the derivative w.r.t a complex variable c=x+iy in the usual fashion:

D = d/dc = (d/dx - id/dy)/2

and

__D__ = d/d__c__ = (d/dx + id/dy)/2

where we use an underline to denote complex conjugate (because 'overline', the usual way of showing this, is not available in html). The harmonic or Laplacian is then just

4 D__D__ = d^{2}/dx^{2} + d^{2}/dy^{2}

Lets review some trivial factoids to make sure we're on the same
page. Since z_{n}(c,__c__) = z_{n}(c) is not a
function of __c__, we have __D__z = D__z__ = 0 . The modulus
is just |z| = sqrt(z__z__), and the derivative of the modulus is
D|z| = __z__ Dz / (2|z|) . We can now combine these to compute
the Laplacian of the the renormalized iteration count:

4 D__D__ m = Dz_{n} __Dz___{n} /
(log 2 |z_{n}|^{2} log^{2} |z_{n}|)

Note that this depends implicitly on the escape radius R, since by definition,
R < |z_{n}|. Thus, it would seem that the harmonic can be made
arbitrarily small, provided the numerator was bounded. Unfortunately, it
is not; not only is it unbounded, but it diverges 'worse than
exponentially' as one approaches the boundary of the M-set.

The Laplacian of the potential is

4 D__D__ f = 4f (log^{2}2 Dm __D__m - log2 D__D__m)

But

Dm = D|z_{n}| / (log2 |z_{n}| log |z_{n}|)

Substituting from the above, we can easily see

D__D__ f = 0

both for finite R and R->oo. For finite R, f has 'small'
discontinuities, which vanish in the limit; thus the harmonic is
non-zero on the discontinuities, but these vanish in the limit, and
f is harmonic everywhere outside the M-set.
To complete this analysis of this
function, we check the asymptotic behavior.
For finite R, we can write f(R) = 2^{-(n+1)} log |z_{n}| .
Thus, for c approaching the boundary of the M-set, n gets infinitely large,
whereas |z_{n}| < R^{2}
and so f=0 on the boundary.
At the other end, for large c, we can approximate
z_{n+1} ~z_{n}^{2} for any n, and so

m ~ 2 - log(log |c|) / log 2

and

f ~ log |c| / 4

for large c.

Now that we have f, we should play a few games. We could draw a picture of f, but that in fact is 'boring'. f gets exponentially flat as it approaches the boundary of the M-set, and thus, it looks like a very blurry low-res picture of the M-set. Boring.

The gradient is more interesting. The gradient 2Df always points 'uphill':

2Df = f __z___{n} Dz_{n} /
|z_{n}|^{2} log |z_{n}|

The picture below shows the modulus of the gradient.
Of course, it resembles a coronal discharge. If one were to cut out
a metal plate in the shape of an M-set, apply a high voltage to it,
it is exactly this that you would see, because it is the areas of high
electric field (gradient) where the coronal ionization will occur.
Maybe make a chic lightning rod for the barn.

The next picture encodes the angle of the landing rays on the M-set.
It is nothing more than the phase-angle of the gradient.
The color coding is such that black=0, blue=small angle, green=pi,
red=2pi. In other words, green points west, and red/black points
east, just as it was in the Soviet era.

This last image shows some landing rays.
A 'landing ray' is just the integral flow along the gradient lines of
the potential; i.e. the path a rock rolling down the hill to the M-set
would take. We've picked 31 evenly-spaced angles at infinity; the red
lines indicate the flow along these paths. The color gradations help
show the behavior of some neighboring rays: i.e. the blue rays are
nearby neighbors of the red rays. The spreading as they 'land' on the
M-set just helps show that rays become radically divergent. The
Mandelbrot interior is an artifact of the algorithm, and are not a part
of the Douady-Hubbard isomorphism.
See the Winding Page for more details
on the derivation, and more pictures.

- John Milnor,
*Periodic Orbits, External Rays and the Mandelbrot Set: An Expository Account*, (postscript, mirror of StonyBrook preprint).

Copyright (c) 2000 Linas Vepstas. All Rights Reserved.

Linas Vepstas December 2000

linas@linas.org

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