Discrete Holomorphic Local Dynamical Systems

210 Tien-Cuong Dinh and Nessim Sibony
of generality, assume that �ψi�DSH ≤ 1. This implies that m := 〈μ,ψ0〉 is bounded.
The invariance of μ and the induction hypothesis imply that
�
�
�〈μ,m(ψ1 ◦ f n1 )...(ψr ◦ f nr )〉−
r
∏
i=0
�
�
〈μ,ψi〉 �
�
�
= �〈μ,mψ1(ψ2 ◦ f n2−n1 )...(ψr ◦ f nr−n1 )〉−m
r
∏
i=1
�
�
〈μ,ψi〉 � ≤ cd −n
for some constant c > 0. In order to get the desired estimate, it is enough to show that
�
�
�〈μ,(ψ0 − m)(ψ1 ◦ f n1 )...(ψr ◦ f nr
�
�
)〉 � ≤ cd −n .
Observe that the operator ( f n ) ∗ acts on Lp (μ) for p ≥ 1 and its norm is bounded by
1. Using the invariance of μ and Hölder’s inequality, we get for p := r + 1
�
�
�〈μ,(ψ0 − m)(ψ1 ◦ f n1 )...(ψr ◦ f nr
�
�
)〉 �
�
�
= �〈μ,Λ n1 (ψ0 − m)ψ1 ...(ψr ◦ f nr−n1
�
�
)〉 �
≤�Λ n1 (ψ0 − m)� L p (μ)�ψ1� L p (μ) ...�ψr ◦ f nr−n1 �L p (μ)
≤ cd −n1 �ψ1� L p (μ) ...�ψr� L p (μ),
for some constant c > 0. Since �ψi� L p (μ) � �ψi�DSH, the previous estimates imply
the result. Note that as in Theorem 1.83, it is enough to assume that ψi is d.s.h. for
i ≤ r − 1andψr is in L p (μ) for some p > 1. ⊓⊔
The mixing of μ implies that for any measurable observable ϕ, the times series
ϕ ◦ f n , behaves like independent random variables with the same distribution.
For example, the dependence of ϕ ◦ f n and ϕ is weak when n is large: if a,b are
real numbers, then the measure of {ϕ ◦ f n ≤ a and ϕ ≤ b} is almost equal to
μ{ϕ ◦ f n ≤ a}μ{ϕ ≤ b}. Indeed, it is equal to
� μ,(1]−∞,a] ◦ ϕ ◦ f n )(1 ]−∞,b] ◦ ϕ) � ,
and when n is large, mixing implies that the last integral is approximatively equal to
〈μ,1 ]−∞,a] ◦ ϕ〉〈μ,1 ]−∞,b] ◦ ϕ〉 = μ{ϕ ≤ a}μ{ϕ ≤ b} = μ{ϕ ◦ f n ≤ a}μ{ϕ ≤ b}.
The estimates on the decay of correlations obtained in the above results, give at
which speed the observables become “almost independent”. We are going to show
that under weak assumptions on the regularity of observables ϕ, the times series
ϕ ◦ f n , satisfies the Central Limit Theorem (CLT for short). We recall the classical
CLT for independent random variables. In what follows, E(·) denotes expectation,
i.e. the mean, of a random variable.
Theorem 1.85. Let (X,F ,ν) be a probability space. Let Z1,Z2,... be independent
identically distributed (i.i.d. for short) random variables with values in R, and of