Böttcher's equation, named after Lucjan Böttcher, is the functional equation
where is a given analytic function with a superattracting fixed point of order at , (that is, in a neighbourhood of ), with n ≥ 2 is a sought function. The logarithm of this functional equation amounts to Schröder's equation.
Solution 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:
This solution is sometimes called: the Böttcher coordinate the Böttcher functionJ. F. Ritt. On the iteration of rational functions . Trans. Amer. Math. Soc. 21 (1920) 348-356. MR 1501149. the Boettcher map. 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 in a neighbourhood of the fixed point to the function . An especially important case is when is a polynomial of degree , and = ∞ . Explicit One can explicitly compute Böttcher coordinates for:math.stackexchange question: explicitly-calculating-greens-function-in-complex-dynamics power maps Chebyshev polynomials
Examples
For the function h and n=2Chaos by Arun V. Holden Princeton University Press, 14 lip 2014 - 334
the Böttcher function F is:
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 ; ; and Douady and Hubbard.;
See also Schröder's equation External ray
References