# Period-n Rabbit Renormalization. "Rabbit's show"

You see below the *J(-1)* set with superattracting period-2
critical orbit. The map *f*_{-1}^{o2} 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.
## "Embedded" Rabbits

*c*_{4} = -1.3125... (in the center of the "secondary"
*(1/2, 1/2)* M-bulb) is periodic point with period *4* .
*f*_{c4}^{o2} is renormalizable (see the picture to the
left). The critical point has period *2* under iterations of
*f*_{c4}^{o2}, 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.
*f*_{c4}^{o4} 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 last the (^{1}/_{2}, ^{1}/_{2},
^{1}/_{2}) Rabbit.

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*updated* 31 Dec 2013