Böttcher's equation explained

F(h(z)) = (F(z)) ^{n} where

• is a given analytic function with a superattracting fixed point of order at, (that is,

h(z) = a + c(z - a) ^{n+O((z-a)} ^{n+1} ) ~, 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:^[1]

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 in a neighbourhood of the fixed point to the function . An especially important case is when is a polynomial of degree, and = ∞ .^[5]

Explicit

One can explicitly compute Böttcher coordinates for:^[6]

• power maps

z\to z ^{d}

• Chebyshev polynomials

Examples

For the function h and n=2^[7]

h(x) = (x ^{2})/(1-2 x ^{2})

the Böttcher function F is:

F(x) = (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^{[8] [9]} and Douady and Hubbard.^[10]

See also

• Schröder's equation
• External ray

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. . 14 . 155 - 234.
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. Trans. Amer. Math. Soc.. Joseph. Ritt. Joseph Ritt. On the iteration of rational functions. 21. 348–356. 3. 10.1090/S0002-9947-1920-1501149-6. 1920. free.
4. Stawiska. Małgorzata. 1307.7778 . Lucjan Emil Böttcher (1872–1937) - The Polish Pioneer of Holomorphic Dynamics. math.HO. November 15, 2013. 9-->.
5. Cowen . C. C. . 10.1007/BF02193043 . Analytic solutions of Böttcher's functional equation in the unit disk . . 24 . 187–194 . 1982 .
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. Book: Early Days in Complex Dynamics: A history of complex dynamics in one variable during 1906–1942 . Alexander . Daniel S. . Iavernaro . Felice . Alessandro . Rosa . 2012 . 978-0-8218-4464-9 .
9. Fatou. P.. Pierre Fatou. Sur les équations fonctionnelles, I. Bulletin de la Société Mathématique de France. 47. 1919. 161 - 271. 47.0921.02. 10.24033/bsmf.998. free.

   Fatou. P.. Pierre Fatou. Sur les équations fonctionnelles, II. Bulletin de la Société Mathématique de France. 48. 1920. 33 - 94. 47.0921.02. 10.24033/bsmf.1003. free. ; Fatou. P.. Pierre Fatou. Sur les équations fonctionnelles, III. Bulletin de la Société Mathématique de France. 48. 1920. 208 - 314. 47.0921.02. 10.24033/bsmf.1008. free.

10. Douady. J.. Hubbard. Étude dynamique de polynômes complexes (première partie). Publ. Math. Orsay. 1984. A.. 2012-01-22. https://web.archive.org/web/20131224233739/http://portail.mathdoc.fr/PMO/afficher_notice.php?id=PMO_1984_A1#. 2013-12-24. dead.

    Douady. J.. Hubbard. Étude dynamique des polynômes convexes (deuxième partie). Publ. Math. Orsay. 1985. A.. 2012-01-22. https://web.archive.org/web/20131224230602/http://portail.mathdoc.fr/PMO/afficher_notice.php?id=PMO_1985_A3#. 2013-12-24. dead.