For $f(z)=a_dz^d+\cdots$, the <Böttcher coordinate> at infinity is the conformal coordinate $\phi_f$ defined near infinity by
$$
\phi_f(f(z))=\phi_f(z)^d,
\qquad
\phi_f(z)\sim a_d^{1/(d-1)}z.
$$
Its modulus has a dynamically natural extension to the entire basin of infinity
$$
A_\infty(f)=\{z:f^n(z)\to\infty\}.
$$

The <escape-rate Green function of a polynomial> is
$$
G_f(z)=\lim_{n\to\infty}\frac1{d^n}\log^+|f^n(z)|.
$$
It is zero on the filled Julia set $K(f)$, positive and harmonic on $A_\infty(f)$, and satisfies $G_f(f(z))=dG_f(z)$. Near infinity,
$$
G_f(z)=\log|\phi_f(z)|,
$$
so $e^{G_f}$ is precisely the extension of $|\phi_f|$ throughout the basin.

Solution

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

{c}
{title2=$\phi_f$}

For a degree-$d$ polynomial $f(z)=a_dz^d+\cdots$, the Böttcher coordinate near infinity satisfies
$$
\phi_f(f(z))=\phi_f(z)^d,
\qquad
\phi_f(z)\sim a_d^{1/(d-1)}z.
$$
Its modulus extends naturally throughout the basin of infinity.


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