Böttcher's equation

From formulasearchengine
Jump to navigation Jump to search

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

where

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 zn. 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

  1. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
  2. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
  3. Template:Cite arXiv
  4. Template:Cite doi
  5. {{#invoke:Citation/CS1|citation |CitationClass=journal }}; {{#invoke:Citation/CS1|citation |CitationClass=journal }}; {{#invoke:Citation/CS1|citation |CitationClass=journal }}
  6. {{#invoke:Citation/CS1|citation |CitationClass=journal }}; {{#invoke:Citation/CS1|citation |CitationClass=journal }}