Discrete Holomorphic Local Dynamical Systems

Uniformisation of Foliations by Curves 119
In other words, and recalling how V was defined, up to normalization W is simply
the analytic subset of D k × X given by the union of all the discs {z}×gz(D), z ∈ V,
and all the annuli {z}× f (z,Ar), z ∈ D k \V.
Lemma 3.3. There exists an embedding W → D k × P, which respects the fibrations
over D k .
Proof. Set Br = { w ∈ P ||w| > r }. By construction, ∂W has a neighbourhood
isomorphic to D k × Ar. We can glue D k × Br to W by identification of that neighbourhood
with D k × Ar ⊂ D k × Br, i.e. by prolonging each annulus Ar to a disc Br.
The result is a new space � W with a fibration �π : � W → D k such that:
(i) �π −1 (z) � P for every z ∈ V ;
(ii) �π −1 (z) � Br for every z ∈ D k \V.
We shall prove that � W (and hence W )embedsintoD k × P (incidentally, note the
common features with the proof of Theorem 2.3).
For every z ∈ V, there exists a unique isomorphism
such that
ϕz : �π −1 (z) → P
ϕz(∞)=0 , ϕ ′ z(∞)=1 , ϕz(r)=∞
where ∞,r ∈ Br ⊂ �π −1 (z) and the derivative at ∞ is computed using the coordinate
1 w .EveryP-fibration is locally trivial, and so this isomorphism ϕz depends
holomorphically on z. Thus we obtain a biholomorphism
Φ : �π −1 (V) → V × P
and we want to prove that Φ extends to the full � W .
By Koebe’s Theorem, the distorsion of ϕz on any compact K ⊂ Br is uniformly
bounded (note that ϕz(Br) ⊂ C). Hence, for every w0 ∈ Br the holomorphic function
z ↦→ ϕz(w0) is bounded on V. Because the complement of V in D k is thin,
by Riemann’s extension theorem this function extends holomorphically to D k .
This permits to extends the above Φ also to fibers over D k \ V . Still by bounded
distorsion, this extension is an embedding of � W into D k × P. ⊓⊔
Now we can finish the proof of the theorem. Thanks to the previous embedding,
we may “fill in” the holes of W and obtain a D-fibration W over D k . Then, by the
Thullen type theorem of Siu [Siu] (and transfinite induction) the map τ : W → X
can be meromorphically extended to W. This is the meromorphic family of discs
which extends f (D k × Ar).
By comparison with the usual “parametrized” Hartogs extension lemma [Iv1],
one could ask if the almost embedding hypothesis in Theorem 3.1 is really indispensable.
In some sense, the answer is yes. Indeed, we may easily construct a fibered
immersion f : D × Ar → D × P ⊂ P × P, f (z,w) =(z, f0(z,w)), such that: (i) for