Böttcher'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:

:F(a)= 0

This solution is sometimes called:

• the Böttcher coordinate
• the Böttcher function
• 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 h(z) in a neighbourhood of the fixed point to the function z**n. An especially important case is when h(z) is a polynomial of degree n, and a = ∞ .

Explicit

One can explicitly compute Böttcher coordinates for:

• power maps z\to z^d
• Chebyshev polynomials

Examples

For the function h and n=2

:h(x)= \frac{x^2}{1 - 2x^2}

the Böttcher function F is:

:F(x)= \frac{x}{1 + x^2}

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.

References

References

1. Böttcher, L. E.. (1904). "The principal laws of convergence of iterates and their application to analysis (in Russian)". Izv. Kazan. Fiz.-Mat. Obshch..
2. [https://www.ams.org/journals/tran/1920-021-03/S0002-9947-1920-1501149-6/S0002-9947-1920-1501149-6.pdf J. F. Ritt. On the iteration of rational functions . Trans. Amer. Math. Soc. 21 (1920) 348-356. MR 1501149.]
3. Ritt, Joseph. (1920). "On the iteration of rational functions". Trans. Amer. Math. Soc..
4. Stawiska, Małgorzata. (November 15, 2013<!--). "Lucjan Emil Böttcher (1872–1937) - The Polish Pioneer of Holomorphic Dynamics".
5. (1982). "Analytic solutions of Böttcher's functional equation in the unit disk". [[Aequationes Mathematicae]].
6. [https://math.stackexchange.com/questions/4220754/explicitly-calculating-greens-function-in-complex-dynamics/4243188#4243188 math.stackexchange question: explicitly-calculating-greens-function-in-complex-dynamics]
7. [https://books.google.com/books?id=SvT_AwAAQBAJ&dq=%22boettcher+function%22&pg=PA49 Chaos by Arun V. Holden Princeton University Press, 14 lip 2014 - 334]
8. (2012). "Early Days in Complex Dynamics: A history of complex dynamics in one variable during 1906–1942". American Mathematical Soc..
9. Fatou, P.. (1919). "Sur les équations fonctionnelles, I". Bulletin de la Société Mathématique de France.
10. (1984). "Étude dynamique de polynômes complexes (première partie)". Publ. Math. Orsay.