Discrete Holomorphic Local Dynamical Systems

244 Tien-Cuong Dinh and Nessim Sibony
Proof. Let ν be the restriction of the Lebesgue measure to U multiplied by a
constant so that �ν� = 1. Define
νn := d −n
t ( f n ) ∗ (ν) and μN := 1 N
N ∑ νn.
n=1
By Theorem 2.11, μN converge to μ. Choose a constant M > 0 such that J ≤ M
on U. For any constant m > 0, define
�
gm(x) := min log M
� �
,m + logM = min log
J(x) M
J(x) ,m′�
with m ′ := m + logM. This is a family of continuous functions which are positive,
bounded on U and which converge to logM/J when m goes to infinity. Define
sN(x) := 1 N−1
N
∑
q=0
gm( f q (x)).
Using the definition of f ∗ on measures, we obtain
〈νN,sN〉 = 1 N−1
N
∑
q=0
= 1 N−1
N
∑
q=0
= 1 N−1
N
∑
q=0
d −N�
N ∗
t ( f ) (ν),gm ◦ f q�
d −N+q�
�
N−q ∗
t ( f ) (ν),gm
〈νN−q,gm〉 = 〈μN,gm〉.
In order to bound 〈μ,logJ〉 from below, we will bound 〈μN,gm〉 from above.
For α > 0, let U α N denote the set of points x ∈ U such that sN(x) > α. Since
sN(x) ≤ m ′ ,wehave
〈μN,gm〉 = 〈νN,sN〉 ≤m ′ νN(U α N )+α(1 − νN(U α N ))
If νN(U α N ) converge to 0 when N → ∞, then
= α +(m ′ − α)νN(U α N ).
〈μ,gm〉 = lim
N→∞ 〈μN,gm〉≤α and hence 〈μ,logM/J〉≤α.
We determine a value of α such that νN(U α N ) tend to 0.
By definition of νN, wehave
νN(U α �
N )=
Uα d
N
−N
t ( f N ) ∗ �
(ν)=
Uα d
N
−N
t
�
N−1
∏ J ◦ f
q=0
q
�
dν.