I've been reading about Bottcher coordinates lately, and I've seen them in fundamental books like Milnor's and Hubbard's. Although I can see the benefit of semiconjugating a map to a nicer map, I'm not sure of actual results that have been proven using Bottcher coordinates or how one would actually go about using them.
Does anyone have any references for results (the more fundamental or easier to understand, the better) that use Bottcher coordinates?
External rays are important in the study of the Mandelbrot set and Julia sets.
I once made a commutative diagram to aid my understanding of how the smoothed escape time and exterior distance estimates are calculated for Julia sets of $z \to z^p + c$, using $\phi_c \approx \operatorname{id}$ near $\infty$: