I was reading about Bottcher coordinates at infinity, and currently have a problem that would be most easily solved by calculating Green's function (more precisely, I would like to know whether several distinct points in the dynamic plane of a polynomial of degree $d$ are in the domain/range of Bottcher coordinates).
Is there a nice way to calculate Green's function, especially at critical points?
For reference, here is Green's function. Let $p(z)$ be a polynomial with degree $d$ and $\ln^+(x)=\max\{0,ln(x)\}.$ Then Green's function $G_p$ for polynomial $p$ is:
$G_p(z):=\lim_{n\to \infty} \frac{1}{d^n} \ln^+|p^{\circ n}|.$
Thanks in advance!
Short answer: no, except in very special cases you won't be able to explicitly compute the Green function (except where it's equal to zero). Like you said, it is however easy to compute numerically; you can also find rigorous bounds, that will enable you to for instance give a rigorous computer-assisted proof that some explicit points are in the basin of infinity.
Cases where you can explicitly compute Böttcher coordinates: power maps $z \mapsto z^d$ and Chebyshev polynomials.