Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 157 / 3 / a / Solution

For $f(z)=a_d{z}^d+\cdots$, the Böttcher coordinate at infinity is the conformal coordinate ${\varphi }_f$ defined near infinity byIts modulus has a dynamically natural extension to the entire basin of infinity

The escape-rate Green function of a polynomial isIt is zero on the filled Julia set $K(f)$, positive and harmonic on $A_{\infty }(f)$, and satisfies $G_f(f(z))=d{G}_f(z)$. Near infinity,so $e^{G_f}$ is precisely the extension of $|{\varphi }_f|$ throughout the basin.