Böttcher's equation

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Böttcher's equation, named after Lucjan Böttcher, is a functional equation :

F(h(z)) = (F(z))^n\,\!

where h has a given analytic function with a superattracting fixed point of order n at a, that is,

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

in a neighbourhood of a, with n ≥ 2; F(z) is sought. The logarithm of this functional equation amounts to Schröder's equation.

Solution[edit]

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.[1] This solution is sometimes called the Böttcher coordinate. (The complete proof was published by Joseph Ritt in 1920,[2] who was unaware of the original formulation.[3])

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 = ∞ .[4]

Applications[edit]

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 [5] and Douady and Hubbard .[6]

See also[edit]

References[edit]

  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. ^ Ritt, Joseph (1920). "On the iteration of rational functions". Trans. Amer. Math. Soc 21 (3): 348–356. doi:10.1090/S0002-9947-1920-1501149-6. 
  3. ^ Stawiska, Małgorzata (November 15, 2013). "Lucjan Emil Böttcher (1872–1937) - The Polish Pioneer of Holomorphic Dynamics". arXiv:1307.7778 [math.HO].
  4. ^ Cowen, C. C. (1982). "Analytic solutions of Böttcher's functional equation in the unit disk". Aequationes Mathematicae 24: 187–194. doi:10.1007/BF02193043. 
  5. ^ Fatou, P. (1919). "Sur les équations fonctionnelles, I". Bulletin de la Société Mathématique de France 47: 161–271. JFM 47.0921.02. ; Fatou, P. (1920). "Sur les équations fonctionnelles, II". Bulletin de la Société Mathématique de France 48: 33–94. JFM 47.0921.02. ; Fatou, P. (1920). "Sur les équations fonctionnelles, III". Bulletin de la Société Mathématique de France 48: 208–314. JFM 47.0921.02. 
  6. ^ Douady, A.; Hubbard, J. (1984). "Étude dynamique de polynômes complexes (première partie)". Publ. Math. Orsay. ; Douady, A.; Hubbard, J. (1985). "Étude dynamique des polynômes convexes (deuxième partie)". Publ. Math. Orsay.