Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 209
for n ≥ 0, ϕ in L p (μ) and ψ d.s.h. Moreover, for 0 ≤ ν ≤ 2 there is a constant c > 0
such that
|〈μ,(ϕ ◦ f n )ψ〉−〈μ,ϕ〉〈μ,ψ〉| ≤ cd −nν/2 �ϕ� L p (μ)�ψ�C ν
for n ≥ 0, ϕ in L p (μ) and ψ of class C ν .
Proof. We prove the first assertion. Observe that the correlations
In(ϕ,ψ) := |〈μ,(ϕ ◦ f n )ψ〉−〈μ,ϕ〉〈μ,ψ〉|
vanish if ψ is constant. Therefore, we can assume that 〈μ,ψ〉 = 0. In which case,
we have
In(ϕ,ψ)=|〈μ,ϕΛ n (ψ)〉|,
where Λ denotes the Perron-Frobenius operator associated to f .
We can also assume that �ψ�DSH ≤ 1. Corollary 1.41 implies that for 1 ≤ q < ∞,
�Λ n (ψ)� L q (μ) ≤ cqd −n
where c > 0 is a constant independent of n,q and ψ. Now,ifq is chosen so that
p −1 + q −1 = 1, we obtain using Hölder’s inequality that
In(ϕ,ψ) ≤�ϕ� L p (μ)�Λ n (ψ)� L q (μ) ≤ cq�ϕ� L p (μ)d −n .
This completes the proof of the first assertion. The second assertion is proved in the
same way using Corollary 1.42. ⊓⊔
Observe that the above estimates imply that for ψ smooth
lim
sup
n→∞
�ϕ�
L2 ≤1
(μ)
In(ϕ,ψ)=0.
Since smooth functions are dense in L 2 (μ), the convergence holds for every ψ in
L 2 (μ) and gives another proof of the K-mixing. The following result [DNS] gives
estimates for the exponential mixing of any order. It can be extended to Hölder
continuous observables using the second assertion in Theorem 1.83.
Theorem 1.84. Let f , d, μ be as in Theorem 1.83 and r ≥ 1 an integer. Then there
is a constant c > 0 such that
�
�
�〈μ,ψ0(ψ1 ◦ f n1 )...(ψr ◦ f nr )〉−
r
∏
i=0
�
�
〈μ,ψi〉 � ≤ cd
for 0 = n0 ≤ n1 ≤···≤nr,n:= min0≤i