Discrete Holomorphic Local Dynamical Systems

196 Tien-Cuong Dinh and Nessim Sibony
We will use the following version of a lemma due to Briend-Duval [BD2]. Their
proof uses the theory of moduli of annuli.
Lemma 1.55. Let g : Δr → P k be a holomorphic map from a disc of center 0 and
of radius r in C. Assume that area(g(Δr)) counted with multiplicity, is smaller than
1/2. Then for any ε > 0 there is a constant λ > 0 independent of g,r such that the
diameter of g(Δ λ r) is smaller than ε � area(g(Δr)).
Proof. Observe that the lemma is an easy consequence of the Cauchy formula if
g has values in a compact set of C k ⊂ P k . In order to reduce the problem to this
case, it is enough to prove that given an ε0 > 0, there is a constant λ0 > 0such
that diam(g(Δ λ0r)) ≤ ε0. Forε0 small enough, we can apply the above case to g
restricted to Δ λ0r.
By hypothesis, the graphs Γg of g in Δr × P k have bounded area. So, according to
Bishop’s theorem [BS], these graphs form a relatively compact family of analytic
sets, that is, the limits of these graphs in the Hausdorff sense, are analytic sets.
Since area(g(Δr)) is bounded by 1/2, the limits have no compact components.
So, they are also graphs and the family of the maps g is compact. We deduce that
diam(g(Δ λ0r)) ≤ ε0 for λ0 small enough. ⊓⊔
Sketch of the proof of Proposition 1.51. The last assertion in the proposition is
deduced from the first one and Proposition 1.46 applied to a generic point in B. We
obtain that �μ ′ − μ�≤2 √ ν for every ν strictly larger than ν(R,a) which implies
the result.
For the first assertion, the idea is to construct inverse branches for many discs
passing through a and then to apply Theorem 1.54 in order to construct inverse
branches on balls. We can assume that ν is smaller than 1. Choose constants δ > 0,
ε > 0 small enough and then a constant κ > 0 large enough; all independent of n.
Fix now the integer n. Recall that �( f n )∗(ωFS)� = d (k−1)n . By Lemmas 1.52 and
1.53, there is a family F ′ ⊂ F and a constant r > 0 such that L (F ′ ) > 1 − δ
and for any Δ in F ′ , the mass of R ∧ [Δ κ 2 r ] is smaller than ν and the mass of
( f n )∗(ωFS) ∧ [Δκr] is smaller than Ad (k−1)n with A > 0.
Claim. For each Δ in F ′ , f n admits at least (1 − 2ν)d kn inverse branches
gi : Δ κ 2 r → Vi with area(Vi) ≤ Aν −1 d −n . The inverse branches gi can be extended
to a neighbourhood of Δ κ 2 r .
Assuming the claim, we complete the proof of the proposition. Let a1,...,al be
the points in f −n (a), with l ≤ d kn ,andF ′ s ⊂ F ′ the family of Δ’s such that one of
the previous inverse branches gi : Δ κ 2 r → Vi passes through as, thatis,Vi contains
as. The above claim implies that ∑L (F ′ s) ≥ (1 − δ)(1 −2ν)d kn . There are at most
d kn terms in this sum. We only consider the family S of the indices s such that
L (F ′ s) ≥ 1 − 3 √ ν.SinceL (F ′ s) ≤ 1foreverys, wehave
#S +(d kn − #S )(1 − 3 √ ν) ≥ ∑L (F ′ s ) ≥ (1 − δ)(1 − 2ν)dkn .
Therefore, since δ is small, we have #S ≥ (1 − √ ν)d kn . For any index s ∈ S
and for Δ in F ′ s , by Lemma 1.55, the corresponding inverse branch on Δκr, which