Fractals/Iterations in the complex plane/r a directions

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Gallery[edit]

Theory[edit]

z + mz^d[edit]

Class of functions : [1]

where :

Simplest subclass :

simplest example :


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

are attracting directions if :

mz+z^d[edit]

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 [2]
  • q is a natural number
  • p is a natural number smaller then q

Repelling and attracting directions [3]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

References[edit]

  1. Local holomorphic dynamics of diffeomorphisms in dimension one by Filippo Bracci
  2. wikipedia : Turn_(geometry)
  3. math.stackexchange : what-is-the-shape-of-parabolic-critical-orbit