Discrete Holomorphic Local Dynamical Systems

184 Tien-Cuong Dinh and Nessim Sibony
is not the case for spaces of smooth forms because of the critical set. Moreover, we
will see that there is a spectral gap for the action of endomorphisms of P k which is
a useful property in the stochastic study of the dynamical system. The first method,
called the dd c -method, was introduced in [DS1] and developped in [DS6]. It can be
extended to Green currents of any bidegree. We show a convergence result for PB
measures ν towards the Green measure. PB measures are diffuse in some sense; we
will study equidistribution of Dirac masses in the next paragraph.
Recall that f is an endomorphism of P k of algebraic degree d ≥ 2. Define the
Perron-Frobenius operator Λ on test functions ϕ by Λ(ϕ) := d −k f∗(ϕ). More
precisely, we have
Λ(ϕ)(z) := d −k ∑
w∈ f −1 (z)
ϕ(w),
where the points in f −1 (z) are counted with multiplicity. The following proposition
is crucial.
Proposition 1.34. The operator Λ :DSH(P k ) → DSH(P k ) is well-defined, bounded
and continuous with respect to the weak topology on DSH(P k ). The operator
�Λ :DSH(P k ) → DSH(P k ) defined by
is contracting and satisfies the estimate
�Λ(ϕ) := Λ(ϕ) −〈ω k FS ,Λ(ϕ)〉
� �Λ(ϕ)�DSH ≤ d −1 �ϕ�DSH.
Proof. We prove the first assertion. Let ϕ be a quasi-p.s.h. function such that
dd c ϕ ≥−ωFS. We show that Λ(ϕ) is d.s.h. Since ϕ is strongly upper semicontinuous,
Λ(ϕ) is strongly upper semi-continuous, see Appendix A.2. If
dd c ϕ = S−ωFS with S positive closed, we have dd c Λ(ϕ)=d −k f∗(S)−d −k f∗(ωFS).
Therefore, if u is a quasi-potential of d −k f∗(ωFS), thenu + Λ(ϕ) is strongly semicontinuous
and is a quasi-potential of d −k f∗(S). So, this function is quasi-p.s.h. We
deduce that Λ(ϕ) is d.s.h., and hence Λ :DSH(P k ) → DSH(P k ) is well-defined.
Observe that Λ : L 1 (P k ) → L 1 (P k ) is continuous. Indeed, if ϕ is in L 1 (P k ),we
have
�Λ(ϕ)� L 1 = 〈ω k FS,d −k | f∗(ϕ)|〉 ≤ 〈ω k FS,d −k f∗(|ϕ|)〉 = d −k 〈 f ∗ (ω k FS),|ϕ|〉 � �ϕ� L 1.
Therefore, Λ :DSH(P k ) → DSH(P k ) is continuous with respect to the weak topology.
This and the estimates below imply that Λ is a bounded operator.
We prove now the last estimate in the proposition. Write dd c ϕ = S + − S − with
S ± positive closed. We have
dd c � Λ(ϕ)=dd c Λ(ϕ)=d −k f∗(S + − S − )=d −k f∗(S + ) − d −k f∗(S − ).
Since � f∗(S ± )� = d k−1 �S ± � and 〈ω k FS , �Λ(ϕ)〉 = 0, we obtain that � �Λ(ϕ)�DSH ≤
d −1 �S ± �. The result follows. ⊓⊔