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Böttcher's equation

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Böttcher's equation, named after Lucjan Böttcher, is the functional equation

Contents

where

  • h is 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 is a sought function.

    • The logarithm of this functional equation amounts to Schröder's equation.

    Solution

    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 = ∞ .

    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

    Böttcher's equation Wikipedia
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