Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 217
(X ′ ,F ′ ,ν ′ ) of this space, i.e. to the space (X ∪ X ′ ,F ∪ F ′ , ν ν′
2 + 2 ), and to the
function equal to ψ on X and on X ′ . ⊓⊔
Lemma 1.96. Let ψ be an observable such that �ψ� L ∞ (ν) ≤ b for some constant
b ≥ 0, and E(ψ|F1)=0.Then
for every λ ≥ 0.
E(e λψ |F1) ≤ e−λ b + eλ b
2
Proof. We consider the desintegration of ν with respect to g.Forν-almost every x ∈
X, there is a positive measure νx on g −1 (x) such that if ϕ is a function in L 1 (ν) then
�
〈ν,ϕ〉 = 〈νx,ϕ〉dν(x).
X
Since ν is g-invariant, we have
�
�
〈ν,ϕ〉 = 〈ν,ϕ ◦ g〉 = 〈νx,ϕ ◦ g〉dν(x)=
X
�νx�ϕ(x)dν(x).
X
Therefore, νx is a probability measure for ν-almost every x. Using also the
invariance of ν, we obtain for ϕ and φ in L 2 (ν) that
We deduce that
�
〈ν,ϕ(φ ◦ g)〉 =
X
�
= 〈νg(x),ϕ〉φ(g(x))dν(x). X
�
〈νx,ϕ(φ ◦ g)〉dν(x)= 〈νx,ϕ〉φ(x)dν(x)
X
E(ϕ|F1)(x)=〈ν g(x),ϕ〉.
So, the hypothesis in the lemma is that 〈νx,ψ〉 = 0forν-almost every x. It suffices
to check that
〈νx,e λψ 〉≤ e−λ b + eλ b
.
2
But this is a consequence of Lemma 1.95 applied to νx instead of ν. ⊓⊔
We continue the proof of Theorem 1.94. Without loss of generality we
can assume that 〈ν,ψ〉 = 0 and |ψ| ≤1. The general idea is to write ψ =
ψ ′ +(ψ ′′ − ψ ′′ ◦ g) for functions ψ ′ and ψ ′′ in L 2 (ν) such that
E(ψ ′ ◦ g n |Fn+1)=0, n ≥ 0.
In the language of probability theory, these identities mean that (ψ ′ ◦ g n )n≥0 is a
reversed martingale difference as in Gordin’s approach, see also [V]. The strategy
is to prove the weak LDT for ψ ′ and for the coboundary ψ ′′ − ψ ′′ ◦ g. Theorem 1.94
is then a consequence of Lemmas 1.99 and 1.101 below.