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 1.328³ = 2.34 times respectevely. Some self-similar periodic points with its period are shown too. In the last figures rotational angle is very close to 120^o, which accounts for the 3-fold rotational symmetry in the picture. In the center of the picture one has 3 lines meeting, and there are numerous nearby points where 6 lines meet. At each of the letter points there is a tiny replica of M.

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 Ms 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 Ms shrink by a factor of λ² therefore nearby miniature Ms shrink faster than the view window, so they eventually disappear.
• There is a fourth feature not visible in these pictures: For preperiodic point c_o , the Julia set J(c_o ) near the point z = c_o looks very much like the Mandelbrot set M near c_o . This is a theorem of Lei, which we will discuss on the next page.

Proof: We will use Newton's approximation to find a root of an equation   f_Cn^on(0) = 0   for periodic point c_n with period n near c_o . If (c_n - c_o) value is small enough, then
    f_Cn^on(0) = f_Co+(Cn-Co)^on(0) = f_Co^on(0) + (c_n - c_o) ^d/_dc f_C^on(0) |_C=Co = 0
(we do not prove that we can use this approximation). For simplicity we will denote
    d_n = ^d/_dc f_C^on(0) |_C=Co .
As c_o is preperiodic with period 1, than f_Co^on(0) = h for large enough n, therefore
    c_n - c_o = - h/d_n .
Since f_C^o(n+1)(0) = [f_C^on(0)]² + c, it follows that for large n
    d_n+1 = 2 h d_n + 1
and
    (c_n - c_o )/(c_n+1 - c_o ) = (h/d_n)/(h/d_n+1) = d_n+1 /d_n = 2h + 1/d_n.
The limit of this as n approaches infinity is 2h as claimed, because d_n gets arbitrarily large for large n.

[1] Douglas C. Ravenel Fractals and computer graphics