Fractals/Iterations in the complex plane/r a directions

GalleryEdit

TheoryEdit

dimension one means here that f maps complex plain to complex plain ( self map )[1]

z + mz^dEdit

Class of functions :[2]

 

where :

  •  

Simplest subclass :

 

simplest example :

 

W say that roots of unity, complex points v on unit circle  

 

are attracting directions if :

 

mz+z^dEdit

 
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:

 

where  :

  •   is a internal angle ( rotation number) in turns [3]
  • d = r+1 is the multiplicity of the fixed point [4]
  • 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

 

Repelling and attracting directions [5] in turns near alfa fixed point for complex quadratic polynomials  

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

ReferencesEdit

  1. Attracting domains of certain maps tangent to the identity - video
  2. Local holomorphic dynamics of diffeomorphisms in dimension one by Filippo Bracci
  3. wikipedia : Turn_(geometry)
  4. Discrete local holomorphic dynamics by Marco Abate
  5. math.stackexchange : what-is-the-shape-of-parabolic-critical-orbit