"Cosine" approximation

Periodic and preperiodic points of the Mandelbrot set can be found by means of polynomials gn(c) = fcon(0) . A few of these polynomials are plotted below near the preperiodic point mo = -2 (the boundary crisis point).
g567
As we know in the vicinity of any preperiodic point the Mandelbrot set is self-similar. Let at the crisis point c = -2 + ε /ρn . H.Hurwitz et al. have shown that for ε /ρn sufficiently small
    gn(-2 + ε /ρn) -> 2 cos(ε 1/2)
where ρn = 4n/6 and n > 1 . A few of the polynomials gn(-2 + ε /ρn) are plotted as a function of ε in Fig.2. As n increases (5, 6, 7...) the polynomials approach the universal function 2 cos(ε 1/2) . The range - in ε - of good approximation rapidly increases with increasing n.
Cosine
The superstable values of c (near -2 for n sufficiently large) are associated with the roots of cos(ε 1/2). We denote the real roots of gn by cn,j where j = 1 corresponds to the root closest to c = -2 with the kneading sequence CLRn-2, j = 2, to the next closest with kneading sequence CLRn-3L and so on
    cn,j = -2 + 6π 2(j - 1/2)2 / 4n .
For n = 25 the cosine approximation yields more then 10,000 real roots closest to c = -2 with an error of less then 1%. One can get an approximate formula for locations of preperiodic points too.

The generalisation of universal scaling constants for these M-set copies are [1]
    δn,j = 16 n/[6π 2 (2j - 1)2] ,     an,j = 4 n/[2π (2j - 1)] .

[1] H.Hurwitz, M.Frame, D.Peak "Scaling symmetries in nonlinear dynamics. A view from parameter space" Physica D 81 (1995) 23.

Tip of the Mandelbrot set

The M-set near the preperiodic point mo = -2 is self-similar under scaling by a factor of |λ| = 4 , therefore every next CLRn (n = 1,2...) orbit is 4 times closer to mo . Corresponding miniature M-set shrinks by a factor of λ2 and eventually disappear. They converge to mo with symbolic dynamics [CL]R where [CL] is preperiodic "tail" of the point and R is its periodic part. The point is the tip of the Mandelbrot set antenna.
Period-n orbit CLRn for large n gets near unstable period-1 orbit, leaves it and eventually returns into the z = 0 point. Period-4 and period-5 orbits CLR2 and CLR3 are shown here. Preperiodic point with the pattern [CLR3]L and c = -1.97393 lies between these two orbits.

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updated 29 October 2002