Discrete Holomorphic Local Dynamical Systems

16 Marco Abate
where β is a formal (and holomorphic) invariant given by
β = 1
�
2πi γ
dz
, (15)
z − f (z)
where the integral is taken over a small positive loop γ about the origin.
Proof. An easy computation shows that if f is given by (14) then(15) holds. Let
us now show that the integral in (15) is a holomorphic invariant. Let ϕ bealocal
biholomorphism fixing the origin, and set F = ϕ −1 ◦ f ◦ ϕ.Then
�
�
1 dz 1
=
2πi γ z − f (z) 2πi ϕ−1 ϕ
◦γ
′ (w)dw
ϕ(w) − f � �
1
� =
ϕ(w) 2πi
ϕ −1 ◦γ
ϕ ′ (w)dw
ϕ(w) − ϕ � � .
F(w)
Now, we can clearly find M, M1 > 0suchthat
�
�
� 1
�
�w
− F(w) −
ϕ ′ (w)
ϕ(w)−ϕ � F(w) �
�
�
�
�
� =
1
�
�ϕ(w)−ϕ � F(w) �� �
�
� ϕ(w)−ϕ
�
�
�
� F(w) �
w − F(w)
|w − F(w)|
≤ M �
�ϕ(w) − ϕ � F(w) �� �
≤ M1,
−ϕ ′ (w)
in a neighbourhood of the origin, where the last inequality follows from the fact
that ϕ ′ (0) �= 0. This means that the two meromorphic functions 1/ � w − F(w) � and
ϕ ′ (w)/ � ϕ(w)−ϕ( � F(w) �� differ by a holomorphic function; so they have the same
integral along any small loop surrounding the origin, and
�
1
2πi γ
dz
z − f (z)
�
1
=
2πi ϕ−1◦γ dw
w − F(w) ,
as claimed.
To prove that f is formally conjugated to g, let us first take a local formal change
of coordinates ϕ of the form
ϕ(z)=z + μz d + Od+1
with μ �= 0, and where we are writing Od+1 instead of O(z d+1 ). It follows that
ϕ −1 (z) =z − μz d + Od+1, (ϕ −1 ) ′ (z) =1 − dμz d−1 + Od and (ϕ −1 ) ( j) = Od− j for
all j ≥ 2. Then using the Taylor expansion of ϕ −1 we get
ϕ −1 ◦ f ◦ ϕ(z)
�
= ϕ −1
ϕ(z)+ ∑ a jϕ(z)
j≥r+1
j
= z +(ϕ −1 ) ′� ϕ(z) �
∑
�
a jz j (1 + μz d−1 + Od) j + Od+2r
j≥r+1
= z +[1− dμz d−1 + Od] ∑ a jz
j≥r+1
j (1 + jμz d−1 + Od)+Od+2r
�
�
�
�
�
(16)
(17)