# # Continuous Dwell

Linas Vepstas “Renormalizing the Mandelbrot Escape” (1997) derives the renormalized continuous escape time:

$\mu = n + 1 - \log_2 \left( \log \left|z\right| \right)$

In the above formula, the value of $$\mu$$ is almost completely independent of the iteration count, and of the escape radius, despite the fact that the right-hand-side of the equation is explicitly dependent on both of these quantities. The renormalized iteration count $$\mu$$ depends only on $$c$$, and is a piecewise-continuous, differentiable function thereof. By using a different analysis, it can be seen that the renormalized iteration count $$\mu$$ is in fact the residue remaining when a pole (due to the infinite sum) is removed. That is, the value of $$\mu$$ closely approximates the result of having iterated to infinity, that is, of having an infinite escape radius, and an infinite maximum iteration count.

## # 1 C99 Code

#include <complex.h>
#include <math.h>

double m_continuous_dwell(int N, double R, double _Complex c)
{
double _Complex z = 0;
for (int n = 0; n < N; ++n)
{
if (cabs(z) > R)
return n + 1 - log2(log(cabs(z)));
z = z * z + c;
}
return -1;
}