Fractals/Iterations in the complex plane/r a directions
< Fractals
Gallery
edit
critical orbits for internal angle from 1/1 to 1/10. True attracting directions

Qth 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
editdimension one means here that f maps complex plain to complex plain ( self map )^{[1]}
z + mz^d
editClass 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^d
editOn the complex zplane ( dynamical plane) there are q directions described by angles:
where :
 is an 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), ... , (2q2)/(2q)  0/q, 1/q, ..., (q1)/q 
References
edit ↑ Attracting domains of certain maps tangent to the identity  video
 ↑ Local holomorphic dynamics of diffeomorphisms in dimension one by Filippo Bracci
 ↑ wikipedia : Turn_(geometry)
 ↑ Discrete local holomorphic dynamics by Marco Abate
 ↑ math.stackexchange : whatistheshapeofparaboliccriticalorbit