By A Grothendieck

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**Example text**

Then every geometrical projection device in D is bounded. Proof. Let us ﬁrst show that d(z, ∂D)/|1 − p˜x(z)| is bounded in D. Indeed we have − 12 log |1 − p˜x (z)| ≤ − 21 log(1 − |˜ px (z)|) ≤ ω 0, p˜x (z) ≤ kD (z0 , z) ≤ c2 − 12 log d(z, ∂D), and thus d(z, ∂D)/|1 − p˜x (z)| ≤ exp(2c2 ) for all z ∈ D. Angular Derivatives in Several Complex Variables 33 To prove K-boundedness of the reciprocal, we ﬁrst of all notice that p˜x is 1-Julia at x. Indeed, lim inf kD (z0 , z) − ω 0, p˜x (z) z→x ≤ lim inf kD ϕx (0), ϕx (ζ) − ω(0, ζ) = 0.

Furthermore, since D is strongly convex, v is complex tangential to ∂D at x and t → ϕx (t) is transversal, there is an euclidean ball B ⊂ Φ−1 (D) of center (t0 , 0) and radius 1 − t0 for a suitable t0 ∈ (0, 1). Now deﬁne ˜ h: B → ∆ by ˜ η) = f Φ(ζ, η) . h(ζ, We remark that ˜ h(ζ, 0) = f ϕx (ζ) and ∂ ˜ h(ζ, 0)/∂ζ = ∂f ϕx (ζ) /∂v. Hence we can write ˜ η) = f ϕx (ζ) + η ∂f ϕx (ζ) + o(|η|). h(ζ, ∂v Set h(ζ, η) = f ϕx (ζ) + 12 η ∂f ϕx (ζ) = f ϕx (ζ) + η(1 − ζ)1/2 g(ζ), ∂v where g(ζ) = 12 (1 − ζ)−1/2 ∂f ϕx (ζ) /∂v.

49–107, 2004. c Springer-Verlag Berlin Heidelberg 2004 50 John Erik Fornæss Lecture 1 Complex Dynamics in Dimension 1. Lecture 2 Fatou sets in higher dimension Lecture 3 Saddle points Lecture 4 Saddle sets Fig. 1. The Lectures Complex dynamics, Figure 2, can be divided in two parts, the Julia theory and the Fatou theory. With the Julia theory the focus is on the Julia set where the dynamics has chaotic features. In the Fatou theory the focus is on the orderly behaviour on the Fatou set. In addition there are sets which mix these two features.