Discrete Holomorphic Local Dynamical Systems

212 Tien-Cuong Dinh and Nessim Sibony
A straighforward computation using the invariance of ν gives that the variance
σ in the above theorem is equal to
σ = lim
n→∞ n −1/2 �ϕ + ···+ ϕ ◦ g n−1 � L 2 (ν) .
When ϕ is orthogonal to all ϕ ◦g n ,wefindthatσ = �ϕ� L 2 (μ) . So, Gordin’s theorem
assumes a weak dependence and concludes that the observables satisfy the central
limit theorem.
Consider now the dynamical system associated to an endomorphism f of P k as
above. Let B denote the Borel σ-algebra on P k and define Bn := f −n (B). Since
the measure μ satisfies f ∗ (μ) =d k μ, the norms �E(·|Bn)� L 2 (μ) can be expressed
in terms of the operator Λ. We have the following lemma.
Lemma 1.87. Let ϕ be an observable in L 2 (μ). Then
for 1 ≤ p ≤ 2.
E(ϕ|Fn)=Λ n (ϕ) ◦ f n
and �E(ϕ|Bn)� L p (μ) = �Λ n (ϕ)� L p (μ),
Proof. We have
� μ,ϕ(ψ ◦ f n ) � = � d −kn ( f n ) ∗ (μ),ϕ(ψ ◦ f n ) � = � μ,d −kn ( f n )∗[ϕ(ψ ◦ f n )] �
= � μ,Λ n (ϕ)ψ � = � μ,[Λ n (ϕ) ◦ f n ][ψ ◦ f n ] � .
This proves the first assertion. The invariance of μ implies that �ψ ◦ f n � L p (μ) =
�ψ� L p (μ). Therefore, the second assertion is a consequence of the first one. ⊓⊔
Gordin’s Theorem 1.86, Corollaries 1.41 and 1.42, applied to q = 2, give the
following result.
Corollary 1.88. Let f be an endomorphism of algebraic degree d ≥ 2 of P k and μ
its equilibrium measure. Let ϕ be a d.s.h. function or a Hölder continuous function
on P k , such that 〈μ,ϕ〉 = 0. Assume that ϕ is not a coboundary. Then ϕ satisfies
the central limit theorem with the variance σ > 0 given by
σ 2 := 〈μ,ϕ 2 〉 + 2 ∑ 〈μ,ϕ(ϕ ◦ f
n≥1
n )〉.
We give an interesting decomposition of the space L2 0 (μ) which shows that Λ,
acts like a “generalized shift”. Recall that L2 0 (μ) is the space of functions ψ ∈ L2 (μ)
such that 〈μ,ψ〉 = 0. Corollary 1.88 can also be deduced from the following result.
Proposition 1.89. Let f be an endomorphism of algebraic degree d ≥ 2 of P k and
μ the corresponding equilibrium measure. Define
Vn := {ψ ∈ L 2 0 (μ), Λ n (ψ)=0}.
Then, we have Vn+1 = Vn ⊕V1 ◦ f n as an orthogonal sum and L 2 0 (μ)=⊕∞ n=0 V1 ◦ f n
as a Hilbert sum. Let ψ = ∑ψn ◦ f n , with ψn ∈ V1, be a function in L 2 0 (μ).