Böttcher's equation
Böttcher's equation is the functional equation
where
The logarithm of this functional equation amounts to Schröder's equation.
The equation is named after Lucjan Böttcher.
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. This solution is sometimes called the Böttcher coordinate. (The complete proof was published by Joseph Ritt in 1920, who was unaware of the original formulation.)
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 = ∞ .
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 and Douady and Hubbard .
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