Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 187
Proposition 1.37. The operator Λ : W ∗ (P k ) → W ∗ (P k ) is well-defined, bounded
and continuous with respect to the weak topology on W ∗ (P k ). The operator
�Λ : W ∗ (P k ) → W ∗ (P k ) defined by
is contracting and satisfies the estimate
�Λ(ϕ) := Λ(ϕ) −〈ω k FS ,Λ(ϕ)〉
� �Λ(ϕ)�W ∗ ≤ d−1/2 �ϕ�W ∗.
Sketch of proof. As in Proposition 1.34,sinceϕis in L1 (Pk ), Λ(ϕ) is also in L1 (Pk )
and the main point here is to estimate ∂ϕ.LetSbe a positive closed (1,1)-current
on Pk such that √ −1∂ϕ∧ ∂ϕ ≤ S. We show that √ −1∂ f∗(ϕ) ∧ ∂ f∗(ϕ) ≤ dk f∗(S),
in particular, the Poincaré differential dΛ(ϕ) of Λ(ϕ) is in L2 (Pk ).
If a is not a critical value of f and U a small neighbourhood of a,thenf−1 (U) is
the union of dk open sets Ui which are sent bi-holomorphically on U.Letgi : U → Ui
be the inverse branches of f .OnU, we obtain using Schwarz’s inequality that
√
−1∂ f∗(ϕ) ∧ ∂ f∗(ϕ) = √ �
−1
∑∂g ∗ i (ϕ)
�
∧
�
∑∂g ∗ i (ϕ)
≤ d k ∑ √ −1∂g ∗ i (ϕ) ∧ ∂g∗i (ϕ)
= d k �√ �
f∗ −1∂ϕ∧ ∂ϕ .
Therefore, we have √ −1∂ f∗(ϕ) ∧ ∂ f∗(ϕ) ≤ d k f∗(S) out of the critical values of f
which is a manifold of real codimension 2.
Recall that f∗(ϕ) is in L 1 (P k ). It is a classical result in Sobolev spaces theory
that an L 1 function whose gradient out of a submanifold of codimension 2 is
in L 2 , is in fact in the Sobolev space W 1,2 (P k ). We deduce that the inequality
√ −1∂ f∗(ϕ) ∧ ∂ f∗(ϕ) ≤ d k f∗(S) holds on P k , because the left hand side term is an
L 1 form and has no mass on critical values of f . Finally, we have
√ −1∂Λ(ϕ) ∧ ∂Λ(ϕ) ≤ d −k f∗(S).
This, together with the identity � f∗(S)� = dk−1�S�, implies that � �Λ(ϕ)�W ∗ ≤
d−1/2�S�. The proposition follows. �
In the rest of this paragraph, we show that the Green measure μ is moderate,
see Appendix A.4. Recall that a positive measure ν on P k is moderate if there are
constants α > 0andc > 0suchthat
�e −αϕ � L 1 (ν) ≤ c
for ϕ quasi-p.s.h. such that ddcϕ ≥−ωFS and 〈ωk FS ,ϕ〉 = 0. Moderate measures are
PB and by linearity, if ν is moderate and D is a bounded set of d.s.h. functions then
there are constants α > 0andc > 0 such that
�e α|ϕ| � L 1 (ν) ≤ c for ϕ ∈ D.
�