Discrete Holomorphic Local Dynamical Systems

220 Tien-Cuong Dinh and Nessim Sibony
Suppose the lemma for N ≥ 1, we prove it for N + 1. Define
We have
WN+1 := g −1 (WN) ∪W1 = g −1 (WN ∪W ′ ).
ν(WN+1) ≤ ν(g −1 (WN)) + ν(W1)=ν(WN)+ν(W1) ≤ c(N + 1)e −δ b 0 .
We will apply Lemma 1.96 to the function ψ ∗ such that ψ ∗ = ψ ′ on X \ W1 and
ψ ∗ = 0onW1. By Lemma 1.97, wehaveE(ψ ∗ |F1)=0sinceW1 is an element of
F1. The choice of W1 gives that |ψ ∗ |≤b. By Lemma 1.96,wehave
It follows that
E(e λψ∗
E(e λψ′
|F1) ≤ e−λ b + eλ b
2
|F1) ≤ e−λ b + eλ b
2
on X for λ ≥ 0.
on X \W1 for λ ≥ 0.
Now, using the fact that WN+1 and eλ SN(ψ ′ ◦g) are F1-measurable, we can write
�
e λ SN+1(ψ ′ �
)
dν = e λψ′
e λ SN(ψ ′ ◦g)
dν
X\WN+1
�
=
X\WN+1
X\WN+1
E(e λψ′
|F1)e λ SN(ψ ′ ◦g)
dν.
Since WN+1 = g −1 (WN) ∪W1, the last integral is bounded by
sup E(e
X\W1
λψ′
�
|F1)
�
≤
≤ 2
X\g −1 (WN)
� �
e λ SN(ψ ′ ◦g) dν
e−λ b + eλ b
e
2 X\WN
λ SN(ψ ′ )
dν
�
e−λ b �N+1 + eλ b
2
where the last inequality follows from the induction hypothesis. So, the lemma
holds for N + 1. ⊓⊔
The following lemma, together with Lemma 1.99, implies Theorem 1.94.
Lemma 1.101. The function ψ ′ satisfies the weak LDT.
Proof. Fix an ε > 0. By Lemma 1.100, we have, for every λ ≥ 0
ν � SN(ψ ′ ) ≥ Nε � �
−λ Nε
≤ ν(WN)+e
X\WN
≤ cNe −δ b 0 −λ Nε
+ 2e
,
�
e λ SN(ψ ′ ) dν
e−λ b �N + eλ b
2
.