z + mz^d

Class of functions :^[2]

${\displaystyle f(z)=z+mz^{k+1}+O(z^{k+2})}$

where :

• ${\displaystyle m\neq 0}$

Simplest subclass :

${\displaystyle f(z)=z+mz^{k+1}}$

simplest example :

${\displaystyle f(z)=z+z^{2}}$

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

${\displaystyle \upsilon \in \partial \mathbb {D} }$

are attracting directions if :

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

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:

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

where  :

• ${\displaystyle {\frac {p}{q}}}$ 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

${\displaystyle 0\leq p<q}$

Repelling and attracting directions ^[5] in turns near alfa fixed point for complex quadratic polynomials ${\displaystyle f_{m}(z)=z^{2}+mz}$