Value to use as center of Mandelbrot Set zoom?

Question

I'm wondering what complex number or numbers is preferable to have at the center of a view of a Mandelbrot Set as the viewing range becomes increasingly smaller in order that the complexity of the set becomes visible. On any point in the set, the view will eventually become a solid region of points that are all within the set(infinite iterations), while for many points that aren't in the set, a similar thing will occur for a region of points not in the set. I'm wondering if there is a way to determine what the center of the view of the Mandelbrot Set will give a meaningful view for arbitrary viewing ranges.

Answer 1

Misiurewicz points, which lie on the boundary of the Mandelbrot set, provide an interesting location to zoom in. There are some illustrations near the end of the Wikipedia article. By definition, Misiurewicz points $M_{k,n}$ are the roots of equation $f_c^{(k)}(0) = f_c^{(k+n)}(0)$ where $f(z) = z^2+c$ and superscipts mean iteration. Two simple examples are $-2$ (which is $M_{2,1}$) and $i$ (which is $M_{2,2}$), but the more interesting points (with larger $k$ or $n$) lead to algebraic equations for $c$ that cannot be solved exactly.