Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 157 / 3 / a

Solution

For

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

, the Böttcher coordinate at infinity is the conformal coordinate

ϕ_{f}

defined near infinity by

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

Its modulus has a dynamically natural extension to the entire basin of infinity

A_{\infty} (f) = {z : f^{n} (z) \to \infty} .

G_{f} (z) = lim_{n \to \infty} \frac{1}{d ^{n}} lo g^{+} ∣ f^{n} (z) ∣.

It is zero on the filled Julia set

K (f)

, positive and harmonic on

A_{\infty} (f)

, and satisfies

G_{f} (f (z)) = d G_{f} (z)

G_{f} (z) = lo g ∣ ϕ_{f} (z) ∣,

e^{G_{f}}

is precisely the extension of

∣ ϕ_{f} ∣

throughout the basin.

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