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
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.
Previous: Cosine approximation
Next: Ordering at the interior crisis points
updated 17 August 2002