Böttcher's equation ( $ϕ (f (z)) = ϕ (z)^{m}$ )

Böttcher's equation asks for a holomorphic coordinate conjugating a map with a superattracting fixed point to a monomial. If

f (z) = a z^{m} + O (z^{m + 1})

with

m \geq 2

and

a \neq = 0

, a normalized local solution has

ϕ (z) = b z + O (z^{2})

with

b^{m - 1} = a

. The Böttcher theorem asserts the existence and uniqueness of this local coordinate after choosing

b

.

Table of contents
- Böttcher theorem Böttcher's equation
  - Böttcher coordinate Böttcher theorem
    - Escape-rate Green function of a polynomial Böttcher coordinate
      - Connected Julia set criterion for a polynomial Escape-rate Green function of a polynomial
    - Mandelbrot set connectedness from the parameter Böttcher coordinate Böttcher coordinate

Böttcher theorem

Near a superattracting fixed point, a holomorphic map is conformally conjugate to its leading monomial.

Böttcher coordinate ( $ϕ_{f}$ )

For a degree-

d

f (z) = a_{d} z^{d} + \dots

, the Böttcher coordinate near infinity satisfies

ϕ_{f} (f (z)) = ϕ_{f} (z)^{d}, ϕ_{f} (z) \sim a_{d}^{1/ (d - 1)} z .

Its modulus extends naturally throughout the basin of infinity.

Escape-rate Green function of a polynomial ( $G_{f}$ )

The escape-rate Green function is

G_{f} (z) = lim_{n \to \infty} d^{- n} lo g^{+} ∣ f^{n} (z) ∣.

It vanishes on the filled Julia set, is positive and harmonic on the basin of infinity, satisfies

G_{f} \circ f = d G_{f}

, and equals

lo g ∣ ϕ_{f} ∣

near infinity and by continuation throughout the basin.

Connected Julia set criterion for a polynomial

The Julia set of a polynomial is connected exactly when every finite critical point belongs to the filled Julia set. In this case the Böttcher coordinate extends conformally over the entire basin of infinity; an escaping critical point is precisely an obstruction to that continuation.

Mandelbrot set connectedness from the parameter Böttcher coordinate

For

f_{c} (z) = z^{2} + c

outside the Mandelbrot set, evaluate the dynamical Böttcher coordinate at the critical value:

Φ (c) = ϕ_{c} (c) .

This parameter Böttcher map is a conformal isomorphism from the complement of the Mandelbrot set to the exterior unit disc, proving that the Mandelbrot set is full and connected.

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Böttcher's equation is a mathematical relationship used in the field of mathematics, particularly relating to complex analysis and dynamical systems. It describes the behavior of iterations of complex functions, particularly in the context of studying sequences of meromorphic functions or rational functions. In a basic sense, Böttcher's theorem provides a criterion or a condition under which a given complex dynamical system can be transformed into a simpler form, often related to the concept of normal forms.

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