Discrete Holomorphic Local Dynamical Systems

14 Marco Abate
As a consequence, the telescopic product ∏k ˜ϕk+1/ ˜ϕk converges uniformly on
compact subsets of ˜Hε (and uniformly on H δ ), and thus the sequence ˜ϕk converges,
uniformly on compact subsets, to a holomorphic function ˜ϕ : ˜Hε → C. Sincewe
have
it follows that
˜ϕk ◦ F(w) =F k+1 (w) − F k (w0)= ˜ϕk+1(w)+F � F k (w0) � − F k (w0)
= ˜ϕk+1(w)+1 + O � |F k (w0)| −1/r� ,
˜ϕ ◦ F(w)= ˜ϕ(w)+1
on ˜Hε. In particular, ˜ϕ is not constant; being the limit of injective functions, by
Hurwitz’s theorem it is injective.
We now prove that the image of ˜ϕ contains a right half-plane. First of all, we
claim that
˜ϕ(w)
lim = 1. (12)
|w|→+∞ w
w∈Hδ Indeed, choose η > 0. Since the convergence of the telescopic product is uniform
on Hδ , we can find k0 ∈ N such that
�
� ˜ϕ(w)
�
− ˜ϕk0
�
(w)
�
�
�
η
w − w0
� <
3
on Hδ .Furthermore,wehave
�
� ˜ϕk0 �
�
(w)
�
�
− 1�
w − w0
� =
�
�
� k0 + ∑
�
�
k0−1
j=0 O(|F j (w)| −1/r )+w0 − Fk0(w0) �
�
�
�
w − w0
� = O(|w|−1 )
on Hδ ; therefore we can find R > 0 such that
� �
�
�
˜ϕ(w) �
� − 1�
w − w0
�
< η
3
as soon as |w| > R in Hδ . Finally, if R is large enough we also have
�
�
�
˜ϕ(w)
� −
w − w0
˜ϕ(w)
�
�
�
w � =
� ��
�
�
�
˜ϕ(w) ��
��
w �
�
η
�w
− w0
��
w0
� <
3 ,
and (12) follows.
Equality (12) clearly implies that ( ˜ϕ(w)−w o )/(w−w o ) → 1as|w|→+∞ in H δ
for any w o ∈ C. But this means that if Rew o is large enough then the difference
between the variation of the argument of ˜ϕ − w o along a suitably small closed circle
around w o and the variation of the argument of w − w o along the same circle will
be less than 2π — and thus it will be zero. Then the argument principle implies
that ˜ϕ − w o and w − w o have the same number of zeroes inside that circle, and thus
w o ∈ ˜ϕ(H δ ), as required.