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Period-n Rabbits Renormalization


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 and period-4 rabbits.

"Embedded" Rabbits

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" (1/3) and (1/2) J-sets.

At laste the (1/2, 1/2, 1/2) Rabbit.


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updated 4 June 2003