Discrete Holomorphic Local Dynamical Systems

190 Tien-Cuong Dinh and Nessim Sibony
with c > 0 independent of q and of ψ. On the other hand, by definition of Λ,wehave
�Λ n (ψ) −〈μ,ψ〉� L q (μ) ≤�Λ n (ψ) −〈μ,ψ〉� L ∞ (μ) ≤ 2�ψ� C 0.
The theory of interpolation between the Banach spaces C 0 and C 2 [T1], applied to
the linear operator ψ ↦→ Λ n (ψ) −〈μ,ψ〉, implies that
�Λ n (ψ) −〈μ,ψ〉�Lq (μ) ≤ Aν2 1−ν/2 [cqd −n ] ν/2 �ψ�C ν ,
for some constant Aν > 0 depending only on ν and on P k . This gives the second
inequality in the corollary.
Recall that if L is a linear continuous functional on the space C 0 of continuous
functions, then we have for every 0 < ν < 2
�L�C
1−ν/2
ν ≤ Aν�L�
C 0 �L� ν/2
C 2
for some constant Aν > 0 independent of L (in our case, the functional is with values
in L q (μ)).
For the first inequality, we have for a fixed constant α > 0 small enough,
� μ,e αd nν/2 |Λ n (ψ)−〈μ,ψ〉| � = ∑ q≥0
1
q!
� μ,|αd nν/2 (Λ n (ψ) −〈μ,ψ〉)| q � ≤ ∑ q≥0
1
q! αq c q q q .
By Stirling’s formula, the last sum converges. The result follows. ⊓⊔
Exercise 1.43. Let ϕ be a smooth function and ϕn as in Theorem 1.35. Show that
we can write ϕn = ϕ + n − ϕ− n with ϕ± n quasi-p.s.h. such that �ϕ± n �DSH � d −n and
dd c ϕ ± n � −d−n ωFS. Prove that ϕn converge pointwise to 0 out of a pluripolar set.
Deduce that if ν is a probability measure with no mass on pluripolar sets, then
d −kn ( f n ) ∗ (ν) converge to μ.
Exercise 1.44. Let DSH0(P k ) be the space of d.s.h. functions ϕ such that 〈μ,ϕ〉 = 0.
Show that DSH0(P k ) is a closed subspace of DSH(P k ), invariant under Λ, andthat
the spectral radius of Λ on this space is equal to 1/d. Note that 1 is an eigenvalue
of Λ on DSH(P k ),so,Λ has a spectral gap on DSH(P k ). Prove a similar result for
W ∗ (P k ).
1.4 Equidistribution of Points
In this paragraph, we show that the preimages of a generic point by f n are equidistributed
with respect to the Green measure μ when n goes to infinity. The proof
splits in two parts. First, we prove that there is a maximal proper algebraic set E
which is totally invariant, then we show that for a �∈E , the preimages of a are
equidistributed. We will also prove that the convex set of probability measures ν,