Discrete Holomorphic Local Dynamical Systems

28 Marco Abate
� �
�
card 0 ≤ j ≤ q �
1
� 4 Ω �
λ (2) ≤ εl ≤ 2k = j
2k
.
20 Then
1
k logδk
1
≤ 2 ∑
ν≥0 2ν log�4Ωλ (2 ν+1 ) −1� 1 1
= 2log4+ 2 ∑ log
ν≥0 2ν Ωλ (2ν+1 ) ,
and we are done. ⊓⊔
The second result we would like to present is Yoccoz’s beautiful proof of the fact
that almost every quadratic polynomial f λ is holomorphically linearizable:
Proposition 4.15. The origin is a Siegel point of f λ (z)=λ z + z 2 for almost every
λ ∈ S 1 .
Proof. (Yoccoz [Y2]) The idea is to study the radius of convergence of the inverse
of the linearization of fλ (z)=λz + z2 when λ ∈ Δ ∗ . Theorem 2.4 says that there is
a unique map ϕλ defined in some neighbourhood of the origin such that ϕ ′ λ (0)=1
and ϕλ ◦ f = λϕλ .Letρλbe the radius of convergence of ϕ−1 λ ;wewanttoprove
that ϕλ is defined in a neighbourhood of the unique critical point −λ /2 offλ ,and
that ρλ = |ϕλ (−λ /2)|.
Let Ωλ ⊂⊂ C be the basin of attraction of the origin, that is the set of z ∈ C
whose orbit converges to the origin. Notice that setting ϕλ (z)=λ −k �
ϕλ fλ (z) � we
can extend ϕλ to the whole of Ωλ . Moreover, since the image of ϕ −1
λ is contained
in Ωλ , which is bounded, necessarily ρλ < +∞.LetUλ = ϕ−1 λ (Δρ ).Sincewehave
λ
(ϕ ′ λ ◦ f ) f ′ = λϕ ′ λ
and ϕλ is invertible in Uλ , the function f cannot have critical points in Uλ .
If z = ϕ−1 (33)
λ (w) ∈ Uλ ,wehavef (z)=ϕ−1(λ w) ∈ ϕ−1
λ λ (Δ |λ |ρ ) ⊂⊂ U
λ λ ; therefore
f (U λ ) ⊆ f (U λ ) ⊂⊂ U λ ⊆ Ω λ ,
which implies that ∂U ⊂ Ωλ .Soϕλ is defined on ∂Uλ , and clearly |ϕλ (z)| = ρλ for
all z ∈ ∂Uλ .
If f had no critical points in ∂Uλ ,(33) would imply that ϕλ has no critical points
in ∂Uλ .Butthenϕλwould be locally invertible in ∂Uλ , and thus ϕ −1
would ex-
λ
tend across ∂Δρ , impossible. Therefore −λ /2 ∈ ∂U λ λ ,and|ϕλ (−λ /2)| = ρλ ,as
claimed.
(Up to here it was classic; let us now start Yoccoz’s argument.) Put η(λ )=
ϕλ (−λ /2). From the proof of Theorem 2.4 one easily sees that ϕλ depends holomorphically
on λ ;soη : Δ ∗ → C is holomorphic. Furthermore, since Ωλ ⊆ Δ2,
Schwarz’s lemma applied to ϕ −1
λ : Δρ → Δ2 yields
λ
1 = |(ϕ −1
λ )′ (0)|≤2/ρ λ,
that is ρ λ ≤ 2. Thus η is bounded, and thus it extends holomorphically to the origin.