Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 197
passes through as, has diameter ≤ εd −n/2 . By Theorem 1.54, f n admits an inverse
branch defined on the ball Br and passing through as, with diameter ≤ d−n/2 .This
implies the result.
Proof of the claim. Let νl denote the mass of Rl ∧ [Δκ 2r ]. Then, ∑νl is the mass of
R∧[Δ κ2r ]. Recall that this mass is smaller than ν. By definition, νld kl is the number
of points in f l (Y ) ∩ Δκ 2r , counted with multiplicity. We only have to consider the
case ν < 1. So, we have ν0 = 0andΔκ2rdoes not intersect Y, the critical values
of f . It follows that Δκ 2r admits dk inverse branches for f . By definition of ν1,there
are at most ν1dk such inverse branches which intersect Y, i.e. the images intersect Y .
So, (1 − ν1)dk of them do not meet Y and the image of such a branch admits dk inverse branches for f . We conclude that Δκ 2r admits at least (1 − ν1)d2k inverse
branches for f 2 . By induction, we construct for f n at least (1 − ν1 −···−νn−1)d kn
inverse branches on Δκ 2r .
Now, observe that the mass of ( f n )∗(ωFS)∧[Δκr] is exactly the area of f −n (Δκr).
We know that it is smaller than Ad (k−1)n . It is not difficult to see that Δκ 2r has at
most νd kn inverse branches with area ≥ Aν−1d −n . This completes the proof. �
End of the proof of Theorem 1.45. Let a be a point out of the exceptional set E defined
in Theorem 1.47 for X = Pk .Fixε > 0 and a constant α > 0 small enough. If
μ ′ is a limit value of d−kn ( f n ) ∗ (δa), it is enough to show that �μ ′ − μ�≤2α + 2ε.
Consider Z := {ν(R,z) > ε} and τ as in Proposition 1.50 for X = Pk .Wehave
τ(a)=0. So, for r large enough we have τr(a) ≤ α. Consider all the negative orbits
O j of order r j ≤ r
�
O j = a
( j) j) j)
−r ,...,a( j −1 ,a( 0
( j) j) j)
j)
j)
with f (a −i−1 )=a( −i and a(
0 = a such that a( −r �∈Z and a(
j −i ∈ Z for i �=r j. Each
orbit is repeated according to its multiplicity. Let Sr denote the family of points
b ∈ f −r (a) such that f i (b) ∈ Z for 0 ≤ i ≤ r. Thenf−r (a) \ Sr consists of the
( j)
preimages of the points a −r j . So, by definition of τr, wehave
d −kr #Sr = τr(a) ≤ α
and
d k(r−r j) −kr −r
= d #( f (a) \ Sr)=1 − τr(a) ≥ 1 − α.
d −kr ∑ j
We have for n ≥ r
d −kn ( f n ) ∗ (δa)= d −kn ∑ ( f
b∈Sr
(n−r) ) ∗ (δb)+d −kn ∑( f
j
(n−r j) ∗
)
�
� �
δ ( j) .
a −r j
Since d −kn ( f n ) ∗ preserves the mass of any measure, the first term in the last sum
is of mass d−kr #Sr = τr(a) ≤ α and the second term is of mass ≥ 1 − α. We apply
( j)
Proposition 1.51 to the Dirac masses at a
of d −kn ( f n ) ∗ (δa) then
−r j . We deduce that if μ′ is a limit value
�μ ′ − μ�≤2α + 2ε.
This completes the proof of the theorem. �