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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).