# The Julia set renormalization

On the picture below the central *U'* region (limited by the blue circle) is
mapped by the *f*_{-1} twice to the left (inside the green circle).
Then the circle is mapped one-to-one (quasi-linear) into the *U*
region (inside the red curve). That is the whole
*f*_{-1}^{o2} map from *U'* to *U* is quadratic-like.

*Controls:* Drag the blue circle to change its radius.
Click mouse + *<Alt>/<Ctrl>* to zoom In/Out.
The inverse *f*_{c}^{-2}(z) map has four branches
(see Iterations of inverse maps)

*±(±(z - c)*^{½}
- c)^{½}.

To get the green circle from the red *U* region we shall take
*-(z - c)*^{½}, therefore the whole inverse quadratic-like
map is

*f*_{c}^{-2}(z) =
±(-(z - c)^{½} - c)^{½}.

Iterations of this map are shown above in the red color.
As since *U'* lies in *U*, therefore iterations
of a point in *U* stay in *U'* forever and we get the renormalized
Julia set homeomorphic to *J(0)* (i.e. a circle).
To the right you see even more impressive the renormalized
*J(-1)* midget inside the *J(-1.306)* set.
Renormalization of the *J(-1.5438)* set below corresponding to the
Misiurewicz band merging point is homeomorphic
to the *J(-2)* set (i.e. the straight *[-2, 2]* segment).
*J(-1.4304)* corresponding to the second band merging point is
renormalized in *J(-1.5438)* (the red midget in the center is
equivalent to the whole first picture).

Contents
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*updated* 30 Dec 2013