# Julia Size

The central feature of a Julia set associated with a baby Mandelbrot set is a circle.

Let $$f_0 = 0, f_1 = c, f_2 = c^2 + c, \ldots$$ be the orbit of the origin.

Consider $$z_0 = \lambda$$ on the boundary of the circle.

Then $$z_1 = c + \lambda^2 = f_1 + h$$.

Write the Taylor expansion about $$f$$:

$$z_p = f_p + \frac{\mathrm{d} f_p}{\mathrm{d} f_1} h + \ldots$$.

If $$c$$ is periodic with period $$p$$, then $$f_p = 0$$ and you can pick $$z_0$$ on the boundary of the circle such that $$z_0 = z_p$$.

This gives $$\lambda = \frac{\mathrm{d} f_p}{\mathrm{d} f_1} \lambda^2$$, that is,

$\lambda = 1 / \frac{\mathrm{d} f_p}{\mathrm{d} f_1}$

# 1 C99 Code

#include <complex.h>

double _Complex m_julia_size(double _Complex c, int p)
{
double _Complex z = c;
double _Complex dz = 1;
for (int q = 1; q < p; ++q)
{
dz = 2 * z * dz;
z = z * z + c;
}
return 1 / dz;
}


# 2 Examples

Circle: $$c = 0$$, $$p = 1$$: $$\lambda = 1$$.

Airplane: $$c = -1.7548\ldots$$, $$p = 3$$: $$\lambda = -0.10753\ldots$$.

Kokopelli: $$c= -0.15652\ldots + i 1.0322\ldots$$, $$p = 4$$: $$\lambda = -0.074812\ldots + i 0.038616\ldots$$, $$|\lambda| = 0.084191\ldots$$.