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


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]:

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 ) -> l = 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
    (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

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updated 22 June 2002