Embedded Cantor-like midgets

Cantor-like Julia sets

Cantor Let x1(c) = 1/2 + (1/4 - c)1/2 is unstable fixed point. In Fig 1. we take c = -2.5 and denote A = -x1 and D = x1 . For real maps iterations of points |x| > x1 diverge to infinity, therefore Julia set J(-2.5) Ì [A,D]. Open interval (B,C) is mapped outside [A,D], thus we throw away these points too. As since intervals [A,B] and [C,D] are mapped onto [A,D], therefore we can continue this process ad infinitum. Thus intervals double every iteration and turn into separate points when number of iterations go to infinity.

You see these Cantor-like set on the (x, c) plane (the first picture) and on the complex z plane below.

Animated Cantor trip.

Cantor-like midgets

To "warm up" look at the "airplane" and J(-2) (i.e. the real [-2, 2] segment) midgets for c in the vicinity of the period-3 M-midget. You see yellow outlines of airplane and line segment in the center of each picture.

Here you can see yellow outline of the Cantor set in the Mandelbrot and Julia midgets (the Cantor-like J(-2.15) set is shown for comparison).

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updated 5 June 03