Universal functions f(z) and constants
a_{m/n} ,
d_{m/n} for period
n-tupling can be obtained from functional equations
f(z) = a f^{ on}
(z /a )
Some of these constants are
|d_{1/3}| = 10.09,
Arg(d_{1/3} ) = +-117.1^{o}
|a_{1/3}| = 3.159,
Arg(a_{1/3} ) = 48.36^{o}
There is a simple approximate formula for d
_{m/n} for big n:
d_{1/3} = 4.6 +-8.981 i
1/a_{1/3} = -0.21056 +-0.23681 i
d_{1/4} = -0.85 +-18.11 i
1/a_{1/4} = -0.095 +-0.2738 i
d_{1/5} = -9.5 +-26.4 i
1/a_{1/5} = -0.03 +-0.281 i
d_{2/5} = 19. +-14.6 i
1/a_{2/5} = -0.196 +-0.11 i
d_{m/n} ~ n^{2}
exp(2p i ^{m}/_{n})
The first two approximate values are 4 and 9 instead of 4.66 and 10.09 thus
accuracy is ~10% for all n. Unfortunately there is not such simple
expresion for a_{m/n}.
You can see the Mandelbrot cactus scaling self-similarity in several ways:
1. in the primary (m/n) M-bulbs 1/n^{2} 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...
Due to an approximate formula
d_{m/n} ~ n^{2} exp(2pi ^{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 f = 2p ^{m}/_{n} with radii r_{m/n} = r/n^{2} 3. repeat (2) for every little circle. |
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 4/5To obtain alternative construction of the Farey tree replace each Farey number by its continued fraction representation
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]