# mathr

## Misiurewicz domain coordinates and size estimates

Earlier today I wrote about atom domain coordinates, and thought about extending it to Misiurewicz domains. By simple analogy, define the Misiurewicz domain coordinate $$G$$ with $$0 \le r \lt q$$ and $$1 \le p$$:

$G(c, p, q, r) = \frac{F^{q + p}(0, c) - F^{q}(0, c)}{F^{r + p}(0, c) - F^{r}(0, c)}$

Calculated similarly to the atom domain size estimate, the Misiurewicz domain size estimate is:

$|h| = \left| \frac{F^{r + p}(0, c) - F^{r}(0, c)}{\frac{\partial}{\partial c}F^{q + p}(0, c) - \frac{\partial}{\partial c}F^{q}(0, c)} \right|$

Like the atom domain coordinate, Newton's method can be used to find a point with a given Misiurewicz domain coordinate. Implementing this is left as an exercise (expect an implementation in my mandelbrot-numerics repository at some point soon).