Discrete Holomorphic Local Dynamical Systems

186 Tien-Cuong Dinh and Nessim Sibony
Proposition 1.34 applied inductively on n implies that �ϕn�DSH ≤ d −n �ϕ�DSH.
Since Λ is bounded, it follows that |cn|≤Ad −n �ϕ�DSH, whereA > 0 is a constant.
The property that ν is PB and the above estimate on ϕn imply that 〈ν,ϕn〉 converge
to 0.
We deduce that 〈d −kn ( f n ) ∗ (ν),ϕ〉 converge to cϕ := ∑n≥0 cn and |cϕ| � �ϕ�DSH.
Therefore, d −kn ( f n ) ∗ (ν) converge to a PB measure μ defined by 〈μ,ϕ〉 := cϕ. The
constant cϕ does not depend on ν, hence the measure μ is independent of ν. The
above convergence implies that μ is totally invariant, i.e. f ∗ (μ) =d k μ. Finally,
since cn depends continuously on the d.s.h. function ϕ, the constant cϕ, whichis
defined by a normally convergent series, depends also continuously on ϕ. It follows
that μ is PC.
We prove now the estimates in the theorem. The total invariance of μ implies
that 〈μ,Λ n (ϕ)〉 = 〈μ,ϕ〉 = cϕ.Ifdd c ϕ = S + − S − with S ± positive closed, we have
dd c Λ n (ϕ)=d −kn ( f n )∗(S + ) − d −kn ( f n )∗(S − ), hence
It follows that
For the second estimate, we have
�Λ n (ϕ) − cϕ�μ ≤ d −kn �( f n )∗(S ± )� = d −n �S ± �.
�Λ n (ϕ) − cϕ�μ ≤ d −n �ϕ�μ.
�Λ n (ϕ) − cϕ�DSH = �ϕn�DSH + ∑ ci.
i≥n
The last sum is clearly bounded by a constant times d −n �ϕ�DSH. This together with
the inequality �ϕn�DSH � d −n �ϕ�DSH implies �Λ n (ϕ) − cϕ�DSH � d −n �ϕ�DSH.
We can also use that ��μ and ��DSH are equivalent, see Proposition A.43.
The last inequality in the theorem is then deduced from the identity
〈d −kn ( f n ) ∗ (ν) − μ,ϕ〉 = 〈ν,Λ n (ϕ) − cϕ〉
and the fact that ν is PB. ⊓⊔
Remark 1.36. In the present case, the dd c -method is quite simple. The function ϕn
is the normalized solution of the equation dd c ψ = dd c Λ n (ϕ). It satisfies automatically
good estimates. The other solutions differ from ϕn by constants. We will see
that for polynomial-like maps, the solutions differ by pluriharmonic functions and
the estimates are less straightforward. In the construction of Green (p, p)-currents
with p > 1, ϕ is replaced with a test form of bidegree (k − p,k − p) and ϕn is a
solution of an appropriate dd c -equation. The constants cn will be replaced with
dd c -closed currents with a control of their cohomology class.
The second construction of the Green measure follows the same lines but we use
the complex Sobolev space W ∗ (P k ) instead of DSH(P k ). We obtain that the Green
measure μ is WPB, see Appendix A.4 for the terminology. We only mention here
the result which replaces Proposition 1.34.