Re: Why do they appear there?

On Feb 13, 1:43 am, Tim Little <t...@xxxxxxxxxxxxxxxxxxxxxxxxxx>
wrote:
On 2008-02-13, mike3 <mike4...@xxxxxxxxx> wrote:

Also, why do the Julia sets have the structures they do? For
example, why does the Julia set for c = -0.80625 + 0.13125i have the
little spirals it has? What rules (if any) govern the structure of
the sets?

It's easier to understand the form of the Julia sets than the
Mandelbrot set, at least for me.  Interpreting the space of complex
parameters as a plane (as is usually done), each iteration corresponds
to a transformation of the plane onto itself.  The julia set for some
parameter is the set of points that never escape to infinity.

Spirals form where that mapping locally looks like a small rotation
and expansion.  When iterating the function, small patches near the
"center" of the spiral expand and rotate out a little.  So any feature
in such a patch that appears at one nearby point will also be present
a little further out and around - exactly the behaviour that defines
the shape of a spiral.


Provided, of course, that the orbit doesn't go to infinity, so it
defines points that are within the Julia set.

After "sufficiently many" iterations, the patch may cover a
macroscopic region, though possibly quite distorted since the map
isn't linear.  


Is that then why, for example, the big spirals seen in, say, J for
c = 0.25968749852 + 0.0010937498711i, look distorted (until
you zoom in close on their centers, in which case they look like
regular spirals)?

If that macroscopic region ends up distant from the
origin, then the 'feature' includes no points of the Julia set and is
pretty bland.  If that patch expands out to cover a region nearer the
origin, then it will probably have some Julia points within it and may
include a (possibly very distorted) copy of the whole set.


Is that therefore why the mini-Julia copies appear when one picks a
point near mini-copies of the Mandelbrot set?
.