Multiple renormalization. Shaggy midgets

In the vicinity of the M₃ midget the map f_c^o3 is renormalizable, therefore for c = A, B, C you see "Cauliflower" J(0.35) midgets at z = 0.

Moreover for c = A the map f_A^o23 is renormalizable too, therefore we see the tiny black circle of the J(0) midget in the left picture and cardioid of the M₂₃ midget on the parameter plane. Note, that filaments are similar. They have z_n = L₃ z_n+1² scaling and 2^k-fold symmetry. Small filaments shrink 2 times every new generation.

At last the point B lies in one more M₂₃ midget and the J(C) set below is pure Cantor dust.

You can find your own examples by the "Renormalization" trips applet.

Note, that global dendrite-like structure of J(A) is determined by the f_c map (compare e.g. J(-1.54369) set).

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updated 31 May 2003