# Theory

## z + mz^d

Class of functions : [1]

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

where :

• $m \ne 0$

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

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:

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

where  :

• $\frac{p}{q}$ is a internal angle ( rotation number) in turns [2]
• q is a natural number
• p is a natural number smaller then q

$0 \le p < q$

Repelling and attracting directions [3]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