Discrete Holomorphic Local Dynamical Systems

72 Eric Bedford
for some constant C. On the other hand, the Jacobian δ is constant, so we have
Vol( f n j W0)=|δ| 2n j Vol(W0).
Since f n jW0 ⊂ V ∩{|y| < e δdn j } the first estimate needs to dominate the second
estimate. This is not possible when n j is large, so we conclude that we must have
v = 0onint(K + ). ⊓⊔
1.6 Böttcher Coordinate
We would like to define a function ϕ + = limn→∞ y 1/dn
n , because if we have such a
function, then we will have ϕ + ◦ f =(ϕ + ) d . In order to make this work, we consider
the product
y 1/dn
n
= y0
� y1
y d 0
� 1/d � y2
By the estimate (2), yn/yd n−1 = 1 + q(x j,y j)
yd j
the following:
Theorem 1.29. The limit
ϕ + (x,y) := y lim
y d 1
n−1
∏ n→∞
j=0
� 1/d 2
···
�
yn
y d n−1
� 1/d n
is close to 1 for (x,y) ∈ V + ,sowehave
�
1 + q(x j,yj)
y d j
� 1
d j
exists uniformly on V + and defines a holomorphic function there. Further, we have
ϕ + ◦ f =(ϕ + ) d .
Theorem 1.30.
(1) If γ is a path in U + which starts at a point of V + ,thenϕ + has an analytic
continuation along γ.
(2) There exists no function ψ which is holomorphic on f −1 V + and which is equal
to ϕ + on V + .
Proof. It is clear from the definitions that G + = log|ϕ + | on V + .NowletG ∗ be a
pluriharmonic conjugate for G + in a neighborhood of the starting point of γ, such
that ϕ + = exp(G + + iG ∗ ). Now we may continue G ∗ pluri-harmonically along γ,
which gives us the desired analytic continuation of ϕ + .
The function ψ ◦ f −1 is defined on V + , and it is equal to ϕ + ◦ f −1 on fV + . Thus
(ψ ◦ f −1 ) d =(ϕ ◦ f −1 ) d = ϕ + on fV + . Thus ϕ ◦ f is a dth root of ϕ + on V + .On
the other hand, by the formula, we see that ϕ + ∼ y for (x,y) ∈ V + for |y| large. Since
y does not have a dth root on {|y| > M}, this is a contradiction. ⊓⊔
.