# Attractor

The attractor ww of a point cc within a hyperbolic component of period pp consists of pp points, each satisfying:

Fp(w,c)=wF^p(w,c) = w

Applying Newton’s method in one complex variable:

wm+1=wm−Fp(wm,c)−wm∂∂zFp(wm,c)−1 w_{m+1} = w_m - \frac{F^p(w_m, c) - w_m} {\frac{\partial}{\partial z}F^p(w_m, c) - 1}

A reasonable starting guess for Newton’s method is w0=Fp(0,c)w_0 = F^p(0, c).

# 1 C99 Code

#include <complex.h>

double _Complex m_attractor
    (double _Complex w0, double _Complex c, int p, int n)
{
    double _Complex w = w0;
    for (int m = 0; m < n; ++m)
    {
        double _Complex z = w;
        double _Complex dz = 1;
        for (int i = 0; i < p; ++i)
        {
            dz = 2 * z * dz;
            z = z * z + c;
        }
        w = w - (z - w) / (dz - 1);
    }
    return w;
}

# 2 Examples

c=0.5ic = 0.5 i, p=1p = 1: w0=0.5iw1=−0.1(2500000000000000…)+0.3(7500000000000000…)iw2=−0.1360(2941176470587…)+0.393(38235294117646…)iw3=−0.136009(77572623132…)+0.393075(72864807383…)iw4=−0.13600982475703(358…)+0.3930756888787(0914…)iw5=−0.13600982475703449(…)+0.39307568887871164(…)i\begin{aligned} w_0 &= & 0.5 i \\ w_1 &= -0.1(2500000000000000\ldots) &+ 0.3(7500000000000000\ldots) i \\ w_2 &= -0.1360(2941176470587\ldots) &+ 0.393(38235294117646\ldots) i \\ w_3 &= -0.136009(77572623132\ldots) &+ 0.393075(72864807383\ldots) i \\ w_4 &= -0.13600982475703(358\ldots) &+ 0.3930756888787(0914\ldots) i \\ w_5 &= -0.13600982475703449(\ldots) &+ 0.39307568887871164(\ldots) i \end{aligned}

c=−1.1+0.1ic = -1.1 + 0.1 i, p=2p = 2: w0=0.1−0.12iw1=0.09(5782435714904968…)−0.08(4585559740811250…)iw2=0.09749(9098252211647…)−0.0836(77122424611575…)iw3=0.097497068(763801931…)−0.0836824188(71189990…)iw4=0.097497068806210202(…)−0.083682418894370822(…)i\begin{aligned} w_0 &= 0.1 &- 0.12 i \\ w_1 &= 0.09(5782435714904968\ldots) &- 0.08(4585559740811250\ldots) i \\ w_2 &= 0.09749(9098252211647\ldots) &- 0.0836(77122424611575\ldots) i \\ w_3 &= 0.097497068(763801931\ldots) &- 0.0836824188(71189990\ldots) i \\ w_4 &= 0.097497068806210202(\ldots) &- 0.083682418894370822(\ldots) i \end{aligned}