Here are illustrations of **M** near some of Misiurewicz points.
The images are zoomed *4, 3* and *1.328 ^{3} = 2.34* times
respectevely. Preperiodic points are in the center of the pictures.
Some self-similar periodic points with its period are shown too.

You see, that preperiodic points explain too spokes symmetry in the largest antenna attached to a primary bulb.

These pictures have next features in common [1]:

- The preperiodic points are not in black regions of
**M**. - They exhibit self-similarity, i.e., they look roughly the same at
shrinking the picture centered at preperiodic point by a factor of
*|l|*and rotating through the angle*Arg(l)*. This becomes more precise as the magnification increases. The rotational angles of the sequences are -23.1256^{o}and 119.553^{o}respectively. This accounts for the slight changes in orientation under successive magnifications in figures. - There is a sequence of miniature
**M**s of decreasing size converging to the point. Each of them has a periodic point in its main cardioid (see the theorem below). When we shrink the picture by a factor of*l*the miniature**M**s shrink by a factor of*l*therefore nearby miniature^{2}**M**s shrink faster than the view window, so they eventually disappear. - There is a fourth feature not visible in these pictures: For preperiodic
point
*c*, the Julia set_{0}*J(c*near the point_{0})*z = c*looks very much like the Mandelbrot set_{0}**M**near*c*. This is a theorem of_{0}*Lei*, which we will discuss on the next page.

where

*Proof*:
We will use Newton's approximation to find a root of an equation
*f _{Cn}^{on}(0) = 0*
for periodic point

(we do not prove that we can use this approximation). For simplicity we will denote

As

Since

and

The limit of this as

You can explore preperiodic point vicinity and find "periods" of miniature
**M**s by
"Animated Julia explorer with Orbits 350+350"
(350+450).

[1] *Douglas C. Ravenel*
Fractals and computer graphics

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