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 )

z + mz^d
Class of functions :


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

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


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

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

where : $$ 0 \le p < q  $$
 * $$ \frac{p}{q}$$ is an 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

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

=References=