Universal functions f(z) and constants
am/n ,
dm/n for period
n-tupling can be obtained from functional equations
f(z) = a f on
(z /a )
Some of these constants are
|d1/3| = 10.09,
Arg(d1/3 ) = +-117.1o
|a1/3| = 3.159,
Arg(a1/3 ) = 48.36o
There is a simple approximate formula for d
m/n for big n:
d1/3 = 4.6 +-8.981 i
1/a1/3 = -0.21056 +-0.23681 i
d1/4 = -0.85 +-18.11 i
1/a1/4 = -0.095 +-0.2738 i
d1/5 = -9.5 +-26.4 i
1/a1/5 = -0.03 +-0.281 i
d2/5 = 19. +-14.6 i
1/a2/5 = -0.196 +-0.11 i
dm/n ~ n2
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 am/n.
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...
| Due to an approximate formula
dm/n ~ n2 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 rm/n = r/n2 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 4/5
To 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]