Scaling and ordering at the boundary crisis point

At the crisis point mo = -2 the Mandelbrot set is self-similar with the scaling factor |lmo| = 4 . Each new view below is zoomed in 4 times. m1 = m1,0 is the chaotic bands merging Misiurewicz point with the pattern [CLR]L. CLRn-2 periodic points cn,1 with periods 3, 4, 5, 6... (corresponding to the largest M-sets in the center of each view) and [CLRk+1]L preperiodic points m1,k converg to c = -2 and make a self-similar pattern. Note that these period-3,4,5... M-sets shrink as lmo2.

But you see more fine structures made of tiny M-sets. E.g. (6, 7a, 8a, 9a) and (7b, 8b, 9b, 10b) sequences in the last view. From the "cosine approximation" formula
    cn,j = -2 + 6p 2(j - 1/2)2 / 4n
one can get
    cn+k,(2^k)j = -2 + 6p 2(j - 1/2k+1)2 / 4n .
For k = 0, 1, 2... this sequence with periods n+k converges (from the left side) to the limit point -2 + 6p 2j2 / 4n. Four orbits of the (n=6, j=1) sequence (6, 7a, 8a, 9a) are shown below.

The next period-10 orbit (which finishes the sequence) is the period doubling of the period-5 orbit. The limit point -2 +6p 2 / 46 = -1.98554 lies within the period-5 M-set.

A lot of these fine structures are visible better in the quartic Mandelbrot set antenna below. They have the same topology as the quadratic map filaments but differ in metric behaviour.

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updated 17 August 2002