Here are illustrations of **M** near some of Misiurewicz points.
Preperiodic points are in the center of the pictures.
The images are zoomed *4, 3* and *1.328 ^{3} = 2.34* times
respectevely. 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
*|λ|*and rotating through the angle*Arg(λ)*. 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*λ*the miniature**M**s shrink by a factor of*λ*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_{o}*J(c*near the point_{o })*z = c*looks very much like the Mandelbrot set_{o}**M**near*c*. This is a theorem of_{o }*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

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

Contents Previous: Periodic and preperiodic points in M Next: M and J-sets similarity. Lei's theorem