Discrete Holomorphic Local Dynamical Systems

218 Tien-Cuong Dinh and Nessim Sibony
Let Λg denote the adjoint of the operator ϕ ↦→ ϕ ◦ g on L 2 (ν). These operators
are of norm 1. The computation in Lemma 1.96 shows that E(ϕ|F1)=Λg(ϕ) ◦ g.
We obtain in the same way that E(ϕ|Fn)=Λ n g (ϕ) ◦ g n .Define
ψ ′′ := −
∞
∑
n=1
Λ n g (ψ), ψ′ := ψ − (ψ ′′ − ψ ′′ ◦ g).
Using the hypotheses in Theorem 1.94, we see that ψ ′ and ψ ′′ are in L 2 (ν) with
norms bounded by some constant. However, we loose the uniform boundedness:
these functions are not necessarily in L ∞ (ν).
Lemma 1.97. We have Λ n g (ψ ′ )=0 for n ≥ 1 and E(ψ ′ ◦ g n |Fm)=0 for m > n ≥ 0.
Proof. Clearly Λg(ψ ′′ ◦ g)=ψ ′′ . We deduce from the definition of ψ ′′ that
Λg(ψ ′ )=Λg(ψ) − Λg(ψ ′′ )+Λg(ψ ′′ ◦ g)=Λg(ψ) − Λg(ψ ′′ )+ψ ′′ = 0.
Hence, Λ n g (ψ′ )=0forn ≥ 1. For every function φ in L 2 (ν), sinceν is invariant,
we have for m > n
〈ν,(ψ ′ ◦ g n )(φ ◦ g m )〉 = 〈ν,ψ ′ (φ ◦ g m−n )〉 = 〈ν,Λ m−n
g (ψ ′ )φ〉 = 0.
It follows that E(ψ ′ ◦ g n |Fm)=0. ⊓⊔
Lemma 1.98. There are constants δ0 > 1 and c > 0 such that
ν{|ψ ′ | > b}≤ce −δ b 0 and ν{|ψ ′′ | > b}≤ce −δ b 0
for any b ≥ 0. In particular, tψ ′ and tψ ′′ are ν-integrable for every t ≥ 0.
Proof. Since ψ ′ := ψ − (ψ ′′ − ψ ′′ ◦ g) and ψ is bounded, it is enough to prove the
estimate on ψ ′′ . Indeed, the invariance of ν implies that ψ ′′ ◦ g satisfies a similar
inequality.
Fix a positive constant δ1 such that 1 < δ 2 1
< δ, whereδ is the constant
in Theorem 1.94. Define ϕ := ∑n≥1 δ 2n
1 |Λ n g (ψ)|. We first show that there is
a constant α > 0 such that ν{ϕ ≥ b} � e −αb for every b ≥ 0. Recall that
E(ψ|Fn) =Λ n g (ψ) ◦ gn . Using the hypothesis of Theorem 1.94, the inequality
∑ 1
2n 2 ≤ 1 and the invariance of ν, we obtain for b ≥ 0
ν{ϕ ≥ b} ≤∑ ν
n≥1
= ∑ ν
n≥1
�
�
|Λ n g (ψ)|≥
−2n
δ1 2n2 �
b
δ n |E(ψ|Fn)|≥ δ nδ −2n
1
2n2 ≤ ∑ ν
n≥1
�
b
�
|E(ψ|Fn)|≥
�
� ∑ exp
n≥1
It follows that ν{ϕ ≥ b} � e −αb for some constant α > 0.
−2n
δ1 2n2 −δ nδ −2n
1
2n2 �
b
�
b
.