# Theory

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

## 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

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

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