Universal period n-tuplings cascade of bifurcations

Universal functions f(z) and constants εm/n , δm/n for period n-tupling can be obtained from functional equations
    f(z) = α f on(z /α )
Some of these constants are
    1/3| = 10.09,   Arg(δ1/3 ) = +-117.1o     1/3| = 3.159,   Arg(α1/3 ) = 48.36o
δ1/3 = 4.6 +-8.981 i 1/α1/3 = -0.21056 +-0.23681 i
δ1/4 = -0.85 +-18.11 i 1/α1/4 = -0.095 +-0.2738 i
δ1/5 = -9.5 +-26.4 i 1/α1/5 = -0.03 +-0.281 i
δ2/5 = 19. +-14.6 i 1/α2/5 = -0.196 +-0.11 i
There is a simple approximate formula for δ m/n for big n:
    δm/n ~ n2 exp(2π i m/n)
The first two approximate values are 4 and 9 instead of 4.66 and 10.09.

You can see the Mandelbrot cactus scaling self-similarity in several ways:
1. in the primary (m/n) M-bulbs 1/n2 scaling for different m, n. The Primary (m/n) bulbs Zoo
2. in the (m/n)k M-bulbs bifurcation cascade for different k (parameter plane scaling). (1/4) M-bulbs cascade
3. in the (m/n)k J-bulbs bifurcation cascade for different k (dynamical plane scaling). (1/3) J-bulbs cascade
But before the demonstrations...

How to grow the "Mandelbrot cactus"

Due to an approximate formula
    δm/n ~ n2 exp(2πi m/n).
One can easy "grow" the Mandelbrot cactus (with ~10% accuracy) as like as a L-system tree:
1. take a primary circle with radius r
2. add to the circle little (m/n) leaves at
    φ = 2π m/n with radii rm/n = r/n2
3. repeat (2) for every little circle.
Tony Dixon pointed me out at works where Feigenbaum's theory have been generalised:
1. Goldberg A.I., Sinai Ya.G., Khanin K.M. Usp.Mat.Nauk 38:2, 159 (1983)
2. Cvitanovic, Myrheim (1983)
3. Cvitanovic, Myrheim "Complex Universality" Comm.Math.Phys. 121, 225-254 (1989)

The Farey tree

An interpolation scheme which organizes rational numbers m/n into self-similar levels of increasing period lengths n is provided by Farey tree, a number theoretical construction based on the observations that somwhere midway between two small denominator fractions (such as 1/2 and 1/3) there is the next smallest denominator fraction (such as 2/5), given by the "Farey mediant" (1+1)/(2+3).

Farey tree are constructed by a simple interpolation rule: given two rationals m/n and m'/n' their Farey mediant is given by
    m" / n" = (m + m') / (n + n')
Starting with the ends of unit interval written as 0/1 and 1/1 the rule generates the Farey tree.

   0/1                                         1/1
F0:                      1/2
                       /     \
                   ---         ---- 
                 /                  \
F1:           1/3                    2/3
             /  \                    /  \
           /      \                /      \
F2:     1/4        2/5          3/5        3/4
       /  \        /  \        /  \        /  \
F3: 1/5   2/7   3/8   3/7   4/7   5/8   5/7   4/5
To obtain alternative construction of the Farey tree replace each Farey number by its continued fraction representation
    m / n = [ p1, p2 ... pk ] = 1/(p1+1/(p2+...+1/ pk ))
with pi - positive integers. The next level Farey tree is obtained by replacing the "last 1" in a continued fraction by either 2 = 1 +1 or 1/2 = 1/(1 + 1)
    [ p1, p2 ... q ] -> [ p1, p2 ... q + 1 ] and [ p1, p2 ... q - 1, 2 ]
The resulting Farey tree is given.
                    1/(1+1)
                  /        \
                -            -
              /                \
        1/(2+1)               1/(1+1/(1+1))
       /   \                      /       \
      /     \                    /         \
1/(3+1)   1/(2+1/(1+1))   1/(1+1/(2+1))   1/(1+1/(1+1/(1+1)))
The continued fraction representation shows explicitly that each branch of the Farey tree is similar to the entire tree and suggests scaling low for the associated universal numbers.
[Cvitanovic]
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updated 18 March 2000