Böttcher's equation, named after Lucjan Böttcher, is the functional equation

${\displaystyle F(h(z))=(F(z))^{n}}$

where

• h is a given analytic function with a superattracting fixed point of order n at a, (that is, ${\displaystyle h(z)=a+c(z-a)^{n}+O((z-a)^{n+1})~,}$ in a neighbourhood of a), with n ≥ 2
• F is a sought function.

The logarithm of this functional equation amounts to Schröder's equation.

Solution of functional equation is a function in implicit form.

Lucian Emil Böttcher sketched a proof in 1904 on the existence of solution: an analytic function F in a neighborhood of the fixed point a, such that:^[1]

${\displaystyle F(a)=0}$

This solution is sometimes called:

• the Böttcher coordinate
• the Böttcher function^[2]
• the Boettcher map.

The complete proof was published by Joseph Ritt in 1920,^[3] who was unaware of the original formulation.^[4]

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 = ∞ .^[5]

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^{[8] [9]} and Douady and Hubbard.^[10]

• Schröder's equation
• External ray