Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 195
Proposition 1.51. Let ν be a strictly positive constant. Let a be a point in P k such
that the Lelong number ν(R,a) of R at a is strictly smaller than ν. Then, there is
a ball B centered at a such that f n admits at least (1 − √ ν)d kn inverse branches
gi : B → Ui where Ui are open sets in P k of diameter ≤ d −n/2 . In particular, if μ ′ is
a limit value of the measures d −kn ( f n ) ∗ (δa) then �μ ′ − μ�≤2 � ν(R,a).
Given a local coordinate system at a, letF denote the family of complex lines
passing through a. For such a line Δ denote by Δr the disc of center a and of
radius r. The family F is parametrized by P k−1 where the probability measure (the
volume form) associated to the Fubini-Study metric is denoted by L .LetBr denote
the ball of center a and of radius r.
Lemma 1.52. Let S be a positive closed (1,1)-current on a neighbourhood of
a. Then for any δ > 0 there is an r > 0 and a family F ′ ⊂ F , such that
L (F ′ ) ≥ 1 − δ and for every Δ in F ′ , the measure S ∧ [Δr] is well-defined
and of mass ≤ ν(S,a)+δ,whereν(S,a) is the Lelong number of S at a.
Proof. Let π : � P k → P k be the blow-up of P k at a and E the exceptional hypersurface.
Then, we can write π ∗ (S)=ν(S,a)[E]+S ′ with S ′ a current having no mass
on E, see Exercise A.39. It is clear that for almost every Δr, the restriction of the
potentials of S to Δr is not identically −∞, so, the measure S ∧ [Δr] is well-defined.
Let �Δr denote the strict transform of Δr by π, i.e. the closure of π −1 (Δr \{a}).
Then, the � Δr define a smooth holomorphic fibration over E. The measure S ∧ [Δr] is
equal to the push-forward of π ∗ (S) ∧ [ �Δr] by π. Observe that π ∗ (S) ∧ [ �Δr] is equal
to S ′ ∧ [ � Δr] plus ν(S,a) times the Dirac mass at � Δr ∩ E. Therefore, we only have to
consider the Δr such that S ′ ∧ [ �Δr] are of mass ≤ δ.
Since S ′ have no mass on E, its mass on π −1 (Br) tends to 0 when r tends to 0.
It follows from Fubini’s theorem that when r is small enough the mass of the slices
S ′ ∧ [ � Δr] is ≤ δ except for a small family of Δ. This proves the lemma. ⊓⊔
Lemma 1.53. Let U be a neighbourhood of Br. Let S be a positive closed (1,1)current
on U. Then, for every δ > 0, there is a family F ′ ⊂ F with L (F ′ ) > 1−δ,
such that for Δ in F ′ , the measure S ∧ [Δr] is well-defined and of mass ≤ A�S�,
where A > 0 is a constant depending on δ but independent of S.
Proof. We can assume that �S� = 1. Let π be as in Lemma 1.52. Then, by continuity
of π ∗ , the mass of π ∗ (S) on π −1 (Br) is bounded by a constant. It is enough to
apply Fubini’s theorem in order to estimate the mass of π ∗ (S) ∧ [ � Δr]. ⊓⊔
Recall the following theorem due to Sibony-Wong [SW].
Theorem 1.54. Let m > 0 be a positive constant. Let F ′ ⊂ F be such that
L (F ′ ) ≥ m and let Σ denote the intersection of the family F ′ with Br. Then
any holomorphic function h on a neighbourhood of Σ can be extended to a holomorphic
function on B λ r where λ > 0 is a constant depending on m but independent
of F ′ and r. Moreover, we have
sup
B λr
|h|≤sup|h|.
Σ