"Inverse" quadratic-like maps

Distorted midgets

Sometimes non-linear terms in the local approximation of fc (outside the central region, see Windows of periodicity scaling) lead to distorted midgets (see M and corresponding J(-1) midgets above). But midgets save its topology until these non-central fc are "locally" one-to-one maps.

Renormalization of fc-2 maps

This takes place while the whole fcon map is quadratic-like (has exactly 2 preimages) in the central U' region (between two minima in the picture below).

Quadratic-like mapping on the complex plane z is shown below for c = -1. The central U' region (limited by the blue circle) is mapped twice to the left (inside the green circle). Then the circle is mapped one-to-one in the U region (inside the red curve). That is the whole f-1o2 map from U' to U is 2 to 1. The last picture illustrates too large U' region. The critical point z = 0 get into the fc(U') region (inside the green circle), i.e. the next map is not one-to-one and the red curve has a self-intersection.

Controls: Drag the blue circle to change R'. Click mouse + <Alt>/<Ctrl> to zoom In/Out.
The inverse fc-2(z) map has four branches (see the picture above and Iterations of inverse maps)
    +-(+-(z - c)1/2 - c)1/2 .
To get the green circle from the red U region we shall take -(z - c)1/2, therefore the whole inverse quadratic-like map is
    fc-2(z) = +-(-(z - c)1/2 - c)1/2.
Iterations of this map are shown to the left above in the red color. As since U' lies in U, therefore all preimages of a point in U stay in U' forever and you see renormalized Julia set homeomorphic to J(0) (i.e. a circle). A magnification of the picture is shown to the left below. To the right you see renormalized J(-1) midget inside the J(-1.306) set.

Renormalization of the J(-1.5438) set 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).

At last J(-1.6) is renormalized in the Cantor-like midget.

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updated 10 June 2003