# Theory

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

## z + mz^d

Class of functions :

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

where :

• $m\neq 0$

Simplest subclass :

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

simplest example :

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

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

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

are attracting directions if :

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

## mz+z^d

On the complex z-plane ( dynamical plane) there are q directions described by angles:

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

where  :

• ${\frac {p}{q}}$  is a internal angle ( rotation number) in turns 
• d = r+1 is the multiplicity of the fixed point 
• 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

$0\leq p

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

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