Period-n Rabbit Renormalization. "Rabbit's show"
You see below the J(-1) set with superattracting period-2
critical orbit. The map f-1o2 is renormalizable
(see the right picture below) and z = 0 is its superattracting fixed
point, therefore the biggest central red bulb is homeomorphic to J(0)
(i.e. a circle).
Next pictures illustrate renormalization of the Douady's rabbit.
c4 = -1.3125... (in the center of the "secondary"
(1/2, 1/2) M-bulb) is periodic point with period 4 .
fc4o2 is renormalizable (see the picture to the
left). The critical point has period 2 under iterations of
fc4o2, therefore you see a small "embedded"
homeomorphic copy of J(-1) in the center of the picture. J(-1)
midgets appear in every bulb of "initial" J(-1) set.
fc4o4 is renormalizable too (see below)
and z = 0 is its superattracting fixed point, therefore the
red bulb in the center of J(c4) is a homeomorphic copy of J(0).
In a similar way one can obtain any (m/n)
bifurcation. E.g. this complex (1/3,
1/2) J-set is constructed of the two "primary"
At last the (1/2, 1/2,
Next: Shaggy midgets
updated 31 Dec 2013