Misiurewicz points and the M-set self-similarity
Douglas C. Ravenel

Here are illustrations of M near some of Misiurewicz points. The images are zoomed 4, 3 and 1.3283 = 2.34 times respectevely. Preperiodic points are in the center of the pictures. Some self-similar periodic points with its period are shown too.
click one of these 3 images to Zoom it in 4 times
Zoom=3
Zoom=2.34 In the last figures rotational angle is very close to 120o, 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 -23.1256o and 119.553o 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 c0 , the Julia set J(c0) near the point z = c0 looks very much like the Mandelbrot set M near c0 . This is a theorem of Lei, which we will discuss on the next page.
Theorem Let co be a preperiodic point with period 1. Let cn denote the nearest periodic point with period n. Then as n approaches infinity
(cn - c0 )/(cn+1 - c0 ) -> λ = 2h
where h is the fixed point of the critical orbit of co .

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

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

[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
updated 22 June 2002