Fractals/Iterations in the complex plane/r a directions

Gallery Theory dimension one means here that f maps complex plain to complex plain ( self map )Attracting domains of certain maps tangent to the identity - video z + mz^d Class of functions :Local holomorphic dynamics of diffeomorphisms in dimension one by Filippo Bracci where : Simplest subclass : simplest example : W say that roots of unity, complex points v on unit circle are attracting directions if : mz+z^d thumbnail|right|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 an internal angle ( rotation number) in turns wikipedia : Turn_(geometry) d = r+1 is the multiplicity of the fixed point Discrete local holomorphic dynamics by Marco Abate 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 math.stackexchange : what-is-the-shape-of-parabolic-critical-orbit in turns near alfa fixed point for complex quadratic polynomials Internal angle Attracting directions Repelling directions 1/2 1/4, 3/4 0/2, 1/21/3 1/6, 3/6, 5/6 0/3, 1/3, 2/31/4 1/8, 3/8, 5/8, 7/8 0/4, 1/4, 2/4, 3/41/51/10, 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, 11/120/6, 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