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).

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updated 30 Dec 2013