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Friday, 23 May 2008 02:54 |
On the left picture we see complex structures inside a J set,
made by preimages of fixed and critical points. Preimages of a point z
are f o(-n)(z) . To find f -1
write out
zn+1 = zn2 + c,
zn = +-(zn+1 - c)1/2 or
fc -1 = +-(z - c)1/2.
Since z 1/2 has two values, the number of preimages
doubles after each iteration.
Remember that, when we iterate f, points outside connected J
diverge (are attracted by the critical point at infinity).
Points inside J go to an attracting cycle.
The J set itself consists of boundary points between these two basins.
Repelling fixed points and orbits belong to the J. They become
attracting if we iterate inverse function f -1, therefore
this map can be used to plot J set.
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Douady's Rabbit, Dendrite and Cantor dust Julia sets illustrate inverse
iterations. In this applet the sign of z1/2 is choosed at
random. Points are gathered at the outer border and inner structure of the
sets is vague. One could use both two values of the inverse map to make clear
Julia sets but then I don't know how to zoom the pictures
(and I failed to color images accurately :)
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Iterations of inverse quadratic maps
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