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] This solution is sometimes called: 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] One can explicitly compute Böttcher coordinates for:[6] For the function h and n=2[7] the Böttcher function F is: 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.