# "Inverse" quadratic-like maps

# Distorted midgets

Sometimes non-linear terms in the local approximation of *f*_{c}
(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
*f*_{c} are "locally" one-to-one maps.
# Renormalization of
*f*_{c}^{-2} maps

This takes place while the whole *f*_{c}^{on} 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*_{-1}^{o2} 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 *f*_{c}(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 *f*_{c}^{-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

*f*_{c}^{-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.

Contents
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*updated* 10 June 2003