Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 213
Then, ψ satisfies the Gordin’s condition, see Theorem 1.86, if and only if the
sum ∑n≥1 n�ψn� 2
L 2 (μ) is finite. Moreover, if ψ is d.s.h. (resp. of class C ν , 0 < ν ≤ 2)
with 〈μ,ψ〉 = 0,then�ψn� L 2 (μ) � d −n (resp. �ψn� L 2 (μ) � d −nν/2 ).
Proof. It is easy to check that V ⊥ n = {θ ◦ f n , θ ∈ L2 (μ)}. LetWn+1denote the
orthogonal complement of Vn in Vn+1. Suppose θ ◦ f n is in Wn+1. Then, Λ(θ)=0.
This gives the first decomposition in the proposition.
For the second decomposition, observe that ⊕∞ n=0V1 ◦ f n is a direct orthogonal
sum. We only have to show that ∪Vn is dense in L2 0 (μ).Letθbe an element in ∩V ⊥ n .
We have to show that θ = 0. For every n, θ = θn ◦ f n for appropriate θn. Hence, θ is
measurable with respect to the σ-algebra B∞ := ∩n≥0Bn. We show that B∞ is the
trivial algebra. Let A be an element of B∞. DefineAn = f n (A). SinceAis in B∞,
1A = 1An ◦ f n and Λ n (1A)=1An . K-mixing implies that Λ n (1A) converges in L2 (μ)
to a constant, see Theorem 1.82.So,1Anconverges to a constant which is necessarily
0 or 1. We deduce that μ(An) converges to 0 or 1. On the other hand, we have
μ(An)=〈μ,1An 〉 = 〈μ,1An ◦ f n 〉 = 〈μ,1A〉 = μ(A).
Therefore, A is of measure 0 or 1. This implies the decomposition of L2 0 (μ).
Suppose now that ψ := ∑ψn ◦ f n with Λ(ψn)=0, is an element of L2 0 (μ). We
have E(ψ|Bn)=∑i≥n ψi ◦ f i .So,
∑ �E(ψ|Bn)�
n≥0
2
L2 (μ) = ∑ (n + 1)�ψn�
n≥0
2
L2 (μ) ,
and ψ satisfies Gordin’s condition if and only if the last sum is finite.
Let ψ be a d.s.h. function with 〈μ,ψ〉 = 0. It follows from Theorem 1.83 that
�E(ψ|Bn)�L 2 (μ) = sup
�ϕ�
L2 ≤1
(μ)
|〈μ,(ϕ ◦ f n )ψ〉| � d −n .
Since ψn ◦ f n = E(ψ|Bn) − E(ψ|Bn+1), the above estimate implies that
�ψn� L 2 (μ) = �ψn ◦ f n � L 2 (μ) � d −n .
The case of C ν observables is proved in the same way.
Observe that if (ψn ◦ f n )n≥0 is the sequence of projections of ψ on the factors of
the direct sum ⊕ ∞ n=0 V1 ◦ f n , then the coordinates of Λ(ψ) are (ψn ◦ f n−1 )n≥1. ⊓⊔
We continue the study with other types of convergence. Let us recall the almost
sure version of the central limit theorem in probability theory. Let Zn be random
variables, identically distributed in L 2 (X,F ,ν), such that E(Zn) =0and
E(Z 2 n)=σ 2 , σ > 0. We say that the almost sure central limit theorem holds if at
ν-almost every point in X, the sequence of measures
1
logN
N
∑
n=1
1
n δ n −1/2 ∑ n−1
i=0 Zi