Discrete Holomorphic Local Dynamical Systems

192 Tien-Cuong Dinh and Nessim Sibony
dimension < dimX, such that g −s (E) ⊂ Eforsomes≥ 1, thenE⊂ EX . Moreover,
there are at most finitely many analytic sets in X which are totally invariant under g.
Since g permutes the irreducible components of X, we can find an integer m ≥ 1
such that g m fixes the components of X.
Lemma 1.48. The topological degree of g m is equal to d mp .Moreprecisely,thereis
a hypersurface Y of X containing sing(X) ∪ g m (sing(X)) such that for x ∈ X out of
Y, the fiber g −m (x) has exactly d mp points.
Proof. Since g m fixes the components of X, we can assume that X is irreducible. It
follows that g m defines a covering over some Zariski dense open set of X. Wewant
to prove that δ, the degree of this covering, is equal to d mp . Consider the positive
measure ( f m ) ∗ (ω p
FS ) ∧ [X]. Since( f m ) ∗ (ω p
FS ) is cohomologous to dmpω p
FS ,this
measure is of mass d mp deg(X). Observe that ( f m )∗ preserves the mass of positive
measures and that we have ( f m )∗[X]=δ[X]. Hence,
d mp deg(X) =�( f m ) ∗ (ω p
FS ) ∧ [X]� = �( f m )∗(( f m ) ∗ (ω p
FS ) ∧ [X])�
= �ω p
FS ∧ ( f m )∗[X]� = δ�ω p
FS
∧ [X]� = δ deg(X).
It follows that δ = d mp . So, we can take for Y, a hypersurface which contains the
ramification values of g m and the set sing(X) ∪ g m (sing(X)). ⊓⊔
Let Y be as above. Observe that if g m (x) �∈Y then g m defines a bi-holomorphic
map between a neighbourhood of x and a neighbourhood of g m (x) in X. Let[Y ]
denote the (k − p + 1,k − p + 1)-current of integration on Y in P k .Since( f mn )∗[Y ]
is a positive closed (k − p + 1,k − p + 1)-current of mass d mn(p−1) deg(Y ), we can
define the following ramification current
R = ∑ Rn := ∑ d
n≥0 n≥0
−mnp ( f mn )∗[Y].
Let ν(R,x) denote the Lelong number of R at x. By Theorem A.14, forc > 0,
Ec := {ν(R,x) ≥ c} is an analytic set of dimension ≤ p −1 contained in X.Observe
that E1 contains Y. WewillseethatRmeasures the obstruction for constructing
good backwards orbits.
For any point x ∈ X let λ ′ n (x) denote the number of distinct orbits
x−n,x−n+1,...,x−1,x0
such that gm (x−i−1)=x−i, x0 = x and x−i ∈ X \Y for 0 ≤ i ≤ n − 1. These are the
“good” orbits. Define λn := d−mpnλ ′ n . The function λn is lower semi-continuous with
respect to the Zariski topology on X. Moreover, by Lemma 1.48,wehave0≤λn≤1 and λn = 1 out of the analytic set ∪n−1 i=0 gmi (Y ). The sequence (λn) decreases to a
function λ , which represents the asymptotic proportion of backwards orbits in X \Y.
Lemma 1.49. There is a constant γ > 0 such that λ ≥ γ on X \ E1.