Renormalization on complex plane

It is evident, that one can apply discussed above "linear" theory to the midgets on complex plane. You see below the period-3 Mandelbrot midget located at c3 = -1.7542 . It is b L32 = 52.5334 times smaller then the main M-set. J(0), Rabbit, Cauliflower (and all the rest Julia) midgets shrink L3 = -9.29887 times and are "placed " in the usual typical points (c3, r, c) of the M3 midget. This midgets scaling is called Renormalization (we will discuss severe theory a little later).

The Julia midgets scaling. The "Airplane" structure

Consider the "Airplane" Julia set J(c3) . As we know (see The Julia sets symmetry) fc3 maps the J(0) midget located at z = 0 (the red circle with radius R below) into a small copy (the green circle to the left with radius R2 ) located at z1 = c3 . As was discussed on the previous page fc3 is approximately linear near the point, therefore next the midget is mapped into the green circle at z2 = c32 + c3 with the scaling factor l1 = 2z1 (the map slope at z1 ). At last the midget is mapped into the original circle at z = 0 with the scaling factor l2 = 2z2 (note also, that any k-midget has two preimages located at +-zk-1 ). Thus we get an equation for R
    R2 l1 l2 = R   or     R = 1/ l1 l2 = 1/L2 = -0.1077

You can test (by mouse clicks), that this value coinsides with the red circle raduis (I've found R ~ 0.105 ). Not only that! One can find all the rest midgets by iteration of the reverse map (but it seems tedious a bit :) These J-midgets make a Cantor- like structure on the Im z = 0 axis.
Is there a scaling rule for the midgets locations?

The J(0) midgets hair scaling

The map fc3o3 moves the segment (0, z2) into (0, z1) one, (0, z3) into (0, z2) and so on. Therefore we get infinite pattern made of shrinking self-similar filaments. Squaring z2 transformation leads to the 2,4,8,16...-fold symmetry of these structures.
Points at z = 0 are scaled as   zn = Lk zn+12 . For rn = |zn| = R + dn and dn << R we get
    R + dn = Lk (R + dn+1)2 ~ Lk R2 + 2Lk Rdn+1 = R + 2dn+1
or   dn = 2dn+1 . I.e. for any J(0)-midgets (independently of Lk !) every new generation of small filaments is 2 times smaller and dense then previous one.

The M4 midget scaling

For the biggest period-4 M-midget L4 = -10.55 - 5.448i,   b = 0.7889 - 0.2754i and m = b L42 = 96.14 + 68.23i. So this copy is reduced |m| = 117.88 times and rotated by Arg(m) = 35.36o.

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updated 29 May 2003