On conformal mapping and iteration of rational functions

Abstract

Let J(f) denote the Julia Fatou set of the rational function f and let G be a simply connected invariant component of its complement. Let h map the unit disk D canformally onto G. We consider the connection betwen the repulsive fixed points ζ∈G of f and the fixed points δ∈∂D of the finite Blaschkc product ϕ=hf h. We show in particular that h has an angular limit ζ=h(δ) at δ and give an inequality relating f″(ζ) and ϕ′(δ₁ where h(δ₁)=ζ.

AMS No:

• 30D05
• 30C35