# Böttcher's equation

Böttcher's equation, named after Lucjan Böttcher, is a functional equation :

$F(h(z)) = (F(z))^n\,\!$

where h has a given analytic function with a superattracting fixed point of order n at a, that is,

$h(z)=a+c(z-a)^n+O((z-a)^{n+1}) ~,$

in a neighbourhood of a, with n ≥ 2; F(z) is sought. The logarithm of this functional equation amounts to Schröder's equation.

## Solution

Lucian Emil Böttcher sketched a proof in 1904 on the existence of an analytic solution F in a neighborhood of the fixed point a, such that F(a) = 0.[1] This solution is sometimes called the Böttcher coordinate. (The complete proof was published by Joseph Ritt in 1920,[2] who was unaware of the original formulation.[3])

Böttcher's coordinate (the logarithm of the Schröder function) conjugates h(z) in a neighbourhood of the fixed point to the function zn. An especially important case is when h(z) is a polynomial of degree n, and a = ∞ .[4]

## Applications

Böttcher's equation plays a fundamental role in the part of holomorphic dynamics which studies iteration of polynomials of one complex variable.

Global properties of the Böttcher coordinate were studied by Fatou [5] and Douady and Hubbard .[6]