Fractals/Iterations in the complex plane/r a directions

Gallery

• directions
• 
  critical orbits for internal angle from 1/1 to 1/10. True attracting directions
• 
  Q-th arm stars for q from 1 to 10. Schematic directions
• 
  True repelling directions : external rays that land on alfa fixed point for internal angle from 1/2 to 1/40
• 
  perturbation of parabolic critical orbit

Theory

dimension one means here that f maps complex plain to complex plain ( self map )

z + mz^d

Class of functions :

$f(z)=z+m{z}^{k+1}+O(z^{k+2})$

where :

• $m\ne 0$

Simplest subclass :

$f(z)=z+m{z}^{k+1}$

simplest example :

$f(z)=z+z^2$

W say that roots of unity, complex points v on unit circle $S^1=\{v:abs(v)=1\}$

$\upsilon \in \partial 𝔻$

are attracting directions if :

$\frac{m}{|m|}{\upsilon }^k=-1$

mz+z^d

Critical orbit and directions for for complex quadratic polynomial and internal angle 1/3

On the complex z-plane ( dynamical plane) there are q directions described by angles:

$arg(z)=2\mathrm{\Pi }\frac{p}{q}$

where  :

• $\frac{p}{q}$ is an internal angle ( rotation number) in turns
• d = r+1 is the multiplicity of the fixed point
• r is the number of attracting petals ( which is equal to the number of repelling petals)
• q is a natural number
• p is a natural number smaller then q

$0\le p<q$

Repelling and attracting directions in turns near alfa fixed point for complex quadratic polynomials $f_m(z)=z^2+mz$

Internal angle	Attracting directions	Repelling directions
1/2	1/4, 3/4	0/2, 1/2
1/3	1/6, 3/6, 5/6	0/3, 1/3, 2/3
1/4	1/8, 3/8, 5/8, 7/8	0/4, 1/4, 2/4, 3/4
1/5	1/10, 3/10, 5/10, 7/10, 9/10	0/5, 1/5, 2/5, 3/5, 4/5
1/6	1/12, 3/12, 5/12, 7/12, 9/12, 11/12	0/6, 1/6, 2/6, 3/6, 4/6, 5/6
-	-	-
1/q	1/(2q), 3/(2q), ... , (2q-2)/(2q)	0/q, 1/q, ..., (q-1)/q