# Böttcher's equation

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

where

- Template:Mvar is a given analytic function with a superattracting fixed point of order Template:Mvar at Template:Mvar, (that is, in a neighbourhood of Template:Mvar), 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.^{[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 Template:Mvar 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 Template:Mvar, and Template:Mvar = ∞ .^{[4]}

## 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
^{[5]} and Douady and Hubbard
.^{[6]}

## See also

## References

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