Böttcher's equation

Remarks

So near a superattracting fixed point of local degree $n$, the map $f$ is conjugate to $w \mapsto w^n$. The idea of the proof: after a linear change of variable one may take $a_n = 1$. Then one sets $\varphi(z) = \lim_{k \to \infty} \big(f^k(z)\big)^{1/n^k}$, choosing the branches of the roots that are tangent to the identity. The limit exists because $f^k(z)$ tends to $0$ very fast, and the correction factors form a convergent infinite product. Example: $f(z) = z^2 / (1 - 2z + 2z^2)$ satisfies the equation with $n = 2$ and $\varphi(z) = z/(1 - z)$. Indeed $\varphi(f(z)) = f(z)/(1 - f(z)) = z^2 / (1 - 2z + z^2) = \varphi(z)^2$. For a monic polynomial $P$ of degree $d \ge 2$, the same result at the superattracting point $\infty$ gives a map $\varphi$ near $\infty$ with $\varphi(z) \sim z$ and $\varphi(P(z)) = \varphi(z)^d$. This is the Böttcher coordinate used to define external rays.

Going deeper

For a student

The equation was studied by Lucjan Böttcher in the early twentieth century. It is the superattracting counterpart of Koenigs linearization, which handles multipliers $\lambda$ with $0 < \abs{\lambda} < 1$. For polynomials the Böttcher coordinate at infinity is the main tool for describing the basin of infinity. When the filled Julia set is connected, it extends to a conformal map from the basin of infinity onto the outside of the closed unit disc.