Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 189
By Theorem 1.35, � Λ(�ϕ) belongs to D. This proves the claim. So, the crucial point
is that Λ is contracting on an appropriate hyperplane.
For ϕ,ψ in L 1 (P k ) we have
� � Λ(ϕ) − � �
Λ(ψ)� L1 ≤
�Λ(|ϕ − ψ|)ω k FS = d 1−k
�
|ϕ − ψ| f ∗ (ω k FS) � �ϕ − ψ� L 1.
So, the map � Λ is Lipschitz with respect to dist L 1. In particular, the map
ϕ ↦→ ϕ − Λ(ϕ) is Lipschitz with respect to this distance. Now, we have for ϕ ∈ D
〈μ,ϕ〉 = lim 〈d
n→∞ −kn ( f n ) ∗ (ω k FS ),ϕ〉 = lim
n→∞ 〈ωk FS ,Λ n (ϕ)〉
= 〈ω k FS,ϕ〉−∑ 〈ω
n≥0
k FS,Λ n (ϕ) − Λ n+1 (ϕ)〉
= 〈ω k FS ,ϕ〉−∑ d
n≥0
−n 〈ω k FS , �Λ n (ϕ − Λ(ϕ))〉.
By Lemma 1.19, the last series defines a function on � D which is Hölder continuous
with respect to dist L 1. Therefore, ϕ ↦→〈μ,ϕ〉 is Hölder continuous on D. ⊓⊔
Remark 1.40. Let fs be a family of endomorphisms of algebraic degree d ≥ 2,
depending holomorphically on a parameter s ∈ Σ. Letμs denote its equilibrium
measure. We get that (s,ϕ) ↦→ μs(ϕ) is Hölder continuous on bounded subsets of
Σ × DSH(P k ).
The following results are useful in the stochastic study of the dynamical system.
Corollary 1.41. Let f , μ and Λ be as above. There are constants c > 0 and α > 0
such that if ψ is d.s.h. with �ψ�DSH ≤ 1,then
� μ,e αd n |Λ n (ψ)−〈μ,ψ〉| � ≤ c and �Λ n (ψ) −〈μ,ψ〉� L q (μ) ≤ cqd −n
for every n ≥ 0 and every 1 ≤ q < +∞.
Proof. By Theorem 1.35, d n (Λ n (ψ) −〈μ,ψ〉) belong to a compact family in
DSH(P k ). The first inequality in the corollary follows from Theorem 1.39. Forthe
second one, we can assume that q is integer and we easily deduce the result from
the first inequality and the inequalities x q ≤ q!e x ≤ q q e x for x ≥ 0. ⊓⊔
Corollary 1.42. Let 0 < ν ≤ 2 be a constant. There are constants c > 0 and α > 0
such that if ψ is a ν-Hölder continuous function with �ψ�C ν ≤ 1,then
� μ,e αd nν/2 |Λ n (ψ)−〈μ,ψ〉| � ≤ c and �Λ n (ψ) −〈μ,ψ〉� L q (μ) ≤ cq ν/2 d −nν/2
for every n ≥ 0 and every 1 ≤ q < +∞.
Proof. By Corollary 1.41,since�·�DSH � �·� C 2,wehave
�Λ n (ψ) −〈μ,ψ〉� L q (μ) ≤ cqd −n �ψ� C 2,