Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 225
We prove now the inequality ht( f ) ≤ lov( f ). Consider an (n,ε)-separated
set F in Pk . For each point a ∈ F , let a (n) denote the corresponding point
(a, f (a),..., f n−1 (a)) in Γn and Ba,n the ball of center a (n) and of radius ε/2 in
(Pk ) n .SinceFis (n,ε)-separated, these balls are disjoint. On the other hand,
by Lelong’s inequality, volume(Γn ∩ Ba,n) ≥ c ′ kε2k , c ′ k > 0. Note that Lelong’s inequality
is stated in the Euclidean metric. We can apply it using a fixed atlas of
Pk and the corresponding product atlas of (Pk ) n , the distortion is bounded. So,
#F ≤ c ′−1
k ε−2kvolume(Γn) and hence,
1 1
log#F ≤
n n log(volume(Γn))
�
1
�
+ O .
n
It follows that ht( f ) ≤ lov( f )=k logd. �
We study the entropy of f on some subsets of P k . The following result is due to
de Thélin and Dinh [DT3, D3].
Theorem 1.112. Let f be a holomorphic endomorphism of P k of algebraic degree
d ≥ 2 and Jp its Julia set of order p, 1 ≤ p ≤ k. If K is a subset of P k such that
K ∩ Jp = ∅,thenht( f ,K) ≤ (p − 1)logd.
Proof. The proof is based on Gromov’s idea as in Theorem 1.108 and on the speed
of convergence towards the Green current. Recall that Jp is the support of the
Green (p, p)-current T p of f . Fix an open neighbourhood W of K such that W ⋐
P k \ supp(T p ). Using the notations in Theorem 1.108, we only have to prove that
lov( f ,W ) := limsup
n→∞
1
n
logvolume(Π −1
0 (W ) ∩Γn) ≤ (p − 1)logd.
It is enough to show that volume(Π −1
0 (W) ∩ Γn) � n k d (p−1)n . As in Theorem
1.108, it is sufficient to check that for 0 ≤ ni ≤ n
�
W
( f n1 ) ∗ (ωFS) ∧ ...∧ ( f n k ) ∗ (ωFS) � d (p−1)n .
To this end, we prove by induction on (r,s), 0≤ r ≤ p and 0 ≤ s ≤ k − p + r,that
�T p−r ∧ ( f n1 ) ∗ (ωFS) ∧ ...∧ ( f ns ) ∗ (ωFS)�Wr,s ≤ cr,sd n(r−1) ,
where Wr,s is a neighbourhood of W and cr,s ≥ 0 is a constant independent of n and
of ni. We obtain the result by taking r = p and s = k.
It is clear that the previous inequality holds when r = 0 and also when s = 0. In
both cases, we can take Wr,s = P k \ supp(T p ) and cr,s = 1. Assume the inequality
for (r − 1,s − 1) and (r,s − 1). LetWr,s be a neighbourhood of W strictly contained
in Wr−1,s−1 and Wr,s−1. Letχ ≥ 0 be a smooth cut-off function with support in
Wr−1,s−1 ∩Wr,s−1 which is equal to 1 on Wr,s. We only have to prove that
�
T p−r ∧ ( f n1 ) ∗ (ωFS) ∧ ...∧ ( f ns ) ∗ (ωFS) ∧ χω k−p+r−s
FS
≤ cr,sd n(r−1) .