THE SELF-EXPONENTIATED FRACTAL

Each point *z* gives rise to an object, or a set
of complex points {*Z _{n}*} arising from the iteration
process

*Z _{n+1}* :=

In analogy to what is done to obtain the Mandelbrot set and many other fractals here we can only be interested the divergence or non-divergence of the iteration.

We chose to study the limit cycle period *k*, defined
as the number of points in the limit cycle, for instance 1 if the iteration
converges to one point, 2 if in the limit of large *n* the iteration
jumps between two points, etc. Value *k*=0 means divergence (e.g.
...10^{1010} obviously diverges).

[In numerical work, some value, e.g., *k*=-10 in
my double precision
Fortran
code, needs to be reserved to numerically unresolved cases where *k*
is too large to be established (40 in my program). In retrospect, a large
positive value would have been better.]

Let us first consider the real axis of the complex plane.
I've said that for large positive numbers we expect divergence. But the
situation is different for large negative numbers. For instance, ...(-10)^(-10)^(-10)
converges to a limit cycle of *k*=3 points, very close [to within
~10^(-10)] to -10, 0, and 1. The full survey of the real axis looks as
follows

There are well known results showing that *k*=1 for
*e^(-e)
< Re(z) < e^(1/e) *. Numerically, that's 0.0659880 < x <
1.4446678. (At least these results are known to the students of my friend
Waldek Paluba from Warsaw University. It's a nice exercise to rederive
them, and to understand why Re(z)=1 is not a point where divergence starts).
Also, the switch to *k*=2 at *Re(z) < e^(-e) *follows from
this analysis. The negative axis, however, presents a complicated (fractal)
situation. The result which will be amplified on a complex plane is that
the *k*=2 region is very small compared with the dominant
*k*=3
region.

The analogous cut along the imaginary axis of the complex
plane shows a somewhat smoother picture, symmetrical with respect to the
real axis. We see that the origin of the plane is surrounded by a small
*k*=2
region.

The figure below shows the results of the low-resolution
Complex World Survey. The vertical coordinate is the *Re(z)* contained
in the range (-5,5), and the horizontal coordinate is the *Im(z)*
contained in the same range. Color coding is the following:

black: *k*=3

dark grey: *k*>3

lighter grey (`binoculars'): *k*=1

almost white: *k*=2

white: *k*=0

Complex exponentiation is a contraction and yields a single object (in accord with Banach contraction principle) in the area shaded light grey, which looks like binoculars. At present resolution, and maybe in reality, that part of the complex plane is a contiguous area. (See also the fig. below).

The vertical axis of the figure is the real axis, and
can be compared to the figure *k(x)* above.

At *k*=3 and higher cycle period, the corresponding
set in the *z* plane is clearly fragmented in a fractal manner.

In particular, the areas where the period is too long
to be established by my program seem fractal. The high-*k* regions
tightly surround the stable *k*=1 basin.

And, finally, it is a joy to look at the areas of divergence of complex exponentiation (white).

A closer look at the figure holding outstretched arms shows it is A SORCERER or SORCERESS, resembling the Egyptian animal gods (but then why is there a cross hanging over her?).

The sorceress has a small ball of period-two region on her head.

A detailed look at the region near the *k*=2 ball
is is taken here
.

At a higher resolution, however, the sorcerer(ess) all but disappears and the picture resembles that of her fortress surrounded by lush palm trees and flowers:

This is a **"self-exponentiated"**
fractal, a cousin of a "self-squared"
Mandelbrot
set. The pictures show the period **k** of the limit cycle (0 =
divergence).

The horizontal axis is Im(z) and extends from -3.5 to 3.5 over 800 pixels.

The vertical axis is Re(z) and extends from -4.5 (up) to 2.5 (down) over 700 pixels.

The upper panel:

white: k=0

black: other k

The lower panel:

white: k=0 (or >40)

black: k=3

grey: other k

A more global view of the fractal, whith Im(z) extending
from -10 to 10 along the horizontal axis, and Re(z) from -5 to 15 on the
vertical one, can be viewed here
.
Black denoted divergence of iteration (*k*=0). The palm trees (trunks
of black alternating with *k*=4) have fractal canopy. Alternatively,
one can view the convergence regions (*k* > 0) as a canopy. Then each
level of the hierarchy of leaves has a cycle number larger by one than
the parent set of leaves (fractal
leaf sprouting)

Here is a color version of the central region.

A peculiar "trangular island" filled with diverging (*k*=0)
and very high-periodicity regions (*k*> 20) is found above the palace
(at *z*=-4.1). It shows up as barely one white pixel on the symmetry
axis in the hi-res survey above.

In fact, it's much much more...

It is studied here.

Last revised Feb 24, 1997