Robert P. Munafo, 1997 Nov 19.
The escape radius is a parameter used in iterating a point under the
Mandelbrot Set and Julia Set functions. It is the radius of a circle on
the Complex Plane used as a boundary to determine when iteration can
stop. The circle is centered at the origin, and has a radius of at
least 2.0. As soon as the iteration yields a value that falls outside
the circle, iteration can stop and it has been determined that the
point that was iterated is not a member of the Mandelbrot Set.
The reason for having an escape radius is to prevent iteration from
going on forever. The Mandelbrot iteration is supposed to terminate
when the iteration becomes unbounded, i.e. goes off to infinity. It is
fairly straightforward to show that if a point ends up outside a radius
of 2.0 then the iteration will go off to infinity.
Here is the proof:
%%% need to translate into Mu Ency terminology, try to simplify a bit,
eliminate terms "induction" and "triangle inequality".
Original USENET article:
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From: mgh3@po.cwru.edu (mike hurley)
Newsgroups: sci.math
Subject: Re: Mandelbrot question
Message-ID:
Date: 26 Apr 93 18:58:27 GMT
References: <1993Apr26.173119.1002@eurom.rhein-main.de>
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In article <1993Apr26.173119.1002@eurom.rhein-main.de>,
hein@eurom.rhein-main.de (Hein Roehrig) wrote:
>
>
> I currently am working on a (high school) report concerning the mandelbrot
> set. In one book (The Emperor's New Mind by Penrose) I learned that the
> series is guaranteed to be unbounded if the absolute value of one of its
> elements surpasses 1+sqrt(2). Can anybody explain me why?
>
> Thanks in advance for any advice, including book tips!
>
> Hein
> ----
> eurom: Free Multiline Unix BBS Home of the FSAG
> Frankfurt/Main,Germany Data: ++49-69-6312934
The sequence generated by f(z) = z^2 + c beginning with 0 begins
0,c,c^2 + c, ...
Call these terms z0, z1, z2, ... .
In fact this sequence will be unbounded if any of its terms
has magnitude larger than 2.
The main idea is that if the magnitude |z| is bigger than both
2 and the magnitude |c|, then |f(z)|/|z| > |z| - 1 > 1, so that
by induction the sequence of magnitudes will grow geometrically.
To establish the inequality, note that
|f(z)|/|z| = |z^2 + c|/|z|
>= (|z|^2-|c|)/|z| [ triangle inequality ]
= |z| - (|c|/|z|)
> |z| - 1 [ |z| > |c| ]
> 1 [ |z| > 2 ]
If |c| <= 2 and |zn| > 2 then certainly
z = zn satisfies these hypotheses, so the sequence is
unbounded.
If |c| > 2 then |c^2+c| >= |c|^2 - |c| = |c|(|c|-1) > |c| > 2
so that z=z2 satisfies the hypotheses of the argument above.
mike hurley mgh3@po.cwru.edu
Acknowledgments
Proof that 2.0 is sufficient: Mike Hurley (mgh3@po.cwru.edu)
From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo. Mu-ency index
This work is licensed under a
Creative Commons Attribution 2.5 License
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© 1996-2008 Robert P. Munafo.s.13