# Size Estimate

MSet e-notes “Windows of periodicity scaling” (2013) presents a method to estimate the size of a hyperbolic component:

Under iterations the critical orbit of period $p$ consecutively cycles through $p$ narrow intervals $S_1 \to S_2 \to \ldots \to S_1$ each of width $s_j$ (we choose $S_1$ to include the critical point $z = 0$). We expand $F^p(z, c)$ for small $z$ (in the narrow central interval $S_1$) and $c$ near its value $c_0$ at superstability of period-$p$ attracting orbit. We see that the $s_j$ are small and the map in the intervals $S_2, S_3, \ldots S_n$ may be regarded as approximately linear; the full quadratic map must be retained for the central interval. One thus obtains $$z_{n+p} \approx L_p (z_n^2 + b (c - c_0))$$ where $L_p = l_2 l_3 \ldots l_p$ is the product of the map slopes, $l_n = 2 z_n$ in $p-1$ noncentral intervals and $b = 1 + l_2^{-1} + (l_2 l_3)^{-1} + ... + L_n^{-1}$. We take $L_n$ at $c = c_0$ and treat it as a constant in narrow window. Introducing $Z = L_n z$ and $C = b L_n^2 (c - c_0)$ we get quadratic map $Z_{n+p} = Z_n^2 + C$. Therefore the window width scales like $(b L_n^2)^{-1}$.

Note that the $p-1$ values of $z$ are all non-zero.

#include <complex.h>

double _Complex m_size_estimate
    (double _Complex c, int p)
{
    double _Complex b = 1;
    double _Complex l = 1;
    double _Complex z = 0;
    for (int i = 1; i < p; ++i)
    {
        z = z * z + c;
        l = 2 * z * l;
        b = b + 1 / l;
    }
    return 1 / (b * l * l);
}