Böttcher coordinate
= Böttcher coordinate
{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.