Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 201
every point is ≤ d − 1, then a version of Lojasiewicz’s theorem implies that f n (B)
contains a ball of radius � e−c(d−1)n, c > 0. Therefore, we have
�
�
e −2kc(d−1)n
e λ dnα �
f n e
(B)
λ dnα k
ωFS ≤
P k eα|u| ω k FS .
This contradicts the above exponential estimate.
In general, by Lemma 1.66 below, f n (B) contains always a ball of radius � e−cdn. So, a slightly stronger version of the above exponential estimate will be enough
to get a contradiction. We may improve this exponential estimate: if the Lelong
numbers of S are small, we can increase the constant α and get a contradiction; if
the Lelong numbers of Sn are small, we replace S with Sn.
The assumption un < −λ dn on f n (B) allows to show that all the limit currents of
the sequence d−n ( f n ) ∗ (S) have Lelong numbers larger than some constant ν > 0. If
S ′ is such a current, there are other currents S ′ n such that S ′ = d−n ( f n ) ∗ (S ′ n). Indeed,
if S ′ is the limit of d−ni( f ni) ∗ (S) one can take S ′ n a limit value of d−ni+n ( f ni−n ) ∗ (S).
Let a be a point such that ν(S ′ n,a) ≥ ν. The assumption on the potentials of
S allows to prove by induction on the dimension of the totally invariant analytic
sets that un converge to 0 on the maximal totally invariant set E .So,a is out of
E . Lemma 1.49 allows to construct many distinct points in f −n (a). The identity
S ′ = d−n ( f n ) ∗ (S ′ n ) implies an estimate from below of the Lelong numbers of S′ on
f −n (a). This holds for every n. Finally, this permits to construct analytic sets of
large degrees on which we have estimates on the Lelong numbers of S ′ . Therefore,
S ′ has a too large self-intersection. This contradicts an inequality of Demailly-Méo
[DE,ME3] and completes the proof. Note that the proof of Demailly-Méo inequality
uses Hörmander’s L2 estimates for the ∂ -equation. �
The following lemma is proved in [DS9]. It also holds for meromorphic maps.
Some earlier versions were given in [FS3] and in terms of Lebesgue measure in
[FJ, G2].
Lemma 1.66. There is a constant c > 0 such that if B is a ball of radius r, 0 < r < 1,
in P k ,then f n (B) contains a ball Bn of radius exp(−cr −2k d n ) for any n ≥ 0.
The ball Bn is centered at f n (an) for some point an ∈ B which is not necessarily
the center of B. The key point in the proof of the lemma is to find a point an with
an estimate from below on the Jacobian of f n at an. Ifu is a quasi-potential of the
current of integration on the critical set, the logarithm of this Jacobian is essentially
the value of u + u ◦ f + ···+ u ◦ f n−1 at an. So, in order to prove the existence of a
point an with a good estimate, it is enough to bound the L 1 norm of the last function.
One easily obtains the result using the operator f ∗ :DSH(P k ) → DSH(P k ) and its
iterates, as it is done for f∗ in Proposition 1.34.
Remark 1.67. Let C denote the convex compact set of totally invariant positive
closed (1,1)-currents of mass 1 on P k ,i.e.currentsS such that f ∗ (S) =dS. Define
an operator ∨ on C .IfS1, S2 are elements of C and u1,u2 their dynamical
quasi-potentials, then ui ≤ 0. Since 〈μ,ui〉 = 0andui are upper semi-continuous,