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
βΛ32 = 52.5334 times smaller then
the main M-set. J(0), Rabbit,
Cauliflower (and all the rest Julia) midgets shrink
Λ3 = -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 λ1 = 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 λ1
λ2 = R or
R = 1/ λ1 λ2 =
1/Λ2 = -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 =
Λk zn+12 .
For rn = |zn| = R + dn and
dn << R we get
R + dn = Λk
(R + dn+1)2 ~ Λk
R2 + 2Λk Rdn+1 =
R + 2dn+1 or dn = 2dn+1 . I.e. for any
J(0)-midgets (independently of
Λk !) every new generation of
small filaments is 2 times smaller and dense then previous one.
The M4 midget
For the biggest period-4 M-midget
Λ4 = -10.55 - 5.448i,
β = 0.7889 - 0.2754i and
m = β Λ42
= 96.14 + 68.23i. So this copy is reduced |m| = 117.88 times
and rotated by Arg(m) = 35.36o.