Discrete Holomorphic Local Dynamical Systems

Discrete Holomorphic Local Dynamical Systems 25
such that h−1 ◦ f ◦ h(z)=λ z; we shall prove that h is actually converging. To do so
it suffices to show that
1
sup
k k log|hk| < ∞. (29)
Since f is holomorphic in a neighbourhood of the origin, there exists a number
M > 0suchthat|ak| ≤M k for k ≥ 2; up to a linear change of coordinates we can
assume that M = 1, that is |al|≤1forallk ≥ 2.
Now, h(λ z)= f � h(z) � yields
Therefore
where
Define inductively
and
∑ (λ
k≥2
k − λ )hkz k �
= ∑ al
l≥2
∑ hmz
m≥1
m
�l . (30)
|hk|≤ε −1
k
∑ |hk1 |···|hkν |,
k1 +···+kν =k
ν≥2
εk = |λ k − λ |.
⎧
⎪⎨ 1 if k = 1 ,
αk =
⎪⎩
∑ αk1 ···αkν if k ≥ 2,
k 1 +···+kν =k
ν≥2
⎧
⎨1
if k = 1 ,
δk =
⎩
ε−1 k max δk1 ···δkν , if k ≥ 2.
k 1 +···+kν =k
ν≥2
Then it is easy to check by induction that
|hk|≤αkδk
for all k ≥ 2. Therefore, to establish (29) it suffices to prove analogous estimates
for αk and δk.
To estimate αk, letα = ∑k≥1 αktk .Wehave
�
α − t = ∑ αkt
k≥2
k = ∑
k≥2
∑ α jt
j≥1
j
� k
= α2
1 − α .
This equation has a unique holomorphic solution vanishing at zero
� �
t + 1
α =
4
1 − 1 − 8t
(1 +t) 2
�
,