Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 179
Theorem 1.22. Let f be an endomorphism of algebraic degree d ≥ 2 of Pk . Then,
the Julia set of order 1 of f , i.e. the support J1 of the Green (1,1)-current T ,
coincides with the Julia set J .TheFatousetFisKobayashi hyperbolic and
hyperbolically embedded in Pk . Moreover, for p ≤ k/2,theJuliasetoforderpoff is connected.
Proof. The sequence ( f n ) is equicontinuous on the Fatou set F and f n are holomorphic,
hence the differential Dfn are locally uniformly bounded on F . Therefore,
( f n ) ∗ (ωFS) are locally uniformly bounded on F . We deduce that d−n ( f n ) ∗ (ωFS)
converge to 0 on F . Hence, T is supported on the Julia set J .
Let F ′ denote the complement of the support of T in Pk . Observe that F ′ is
invariant under f n and that −g is a smooth function which is strictly p.s.h. on F ′ .
Therefore, by Proposition 1.21, F ′ is Kobayashi hyperbolic and hyperbolically
embedded in Pk . Therefore, the maps f n , which are self-maps of F ′ , are equicontinuous
with respect to the Kobayashi-Royden metric. On the other hand, the fact
that F ′ is hyperbolically embedded implies that the Kobayashi-Royden metric is
bounded from below by a constant times the Fubini-Study metric. It follows that
( f n ) is locally equicontinuous on F ′ with respect to the Fubini-Study metric. We
conclude that F ′ ⊂ F , hence F = F ′ and J = supp(T )=J1.
In order to show that Jp are connected, it is enough to prove that if S is a
positive closed current of bidegree (p, p) with p ≤ k/2 then the support of S is connected.
Assume that the support of S is not connected, then we can write S = S1 +S2
with S1 and S2 non-zero, positive closed with disjoint supports. Using a convolution
on the automorphism group of Pk , we can construct smooth positive closed
(p, p)-forms S ′ 1 ,S′ 2 with disjoint supports. So, we have S′ 1 ∧ S′ 2 = 0. This contradicts
that the cup-product of the classes [S ′ 1 ] and [S′ 2 ] is non-zero in H2p,2p (Pk ,R) � R:
we have [S ′ 1 ]=�S′ p
1�[ω �[ω p
FS ], [S′ 2 ]=�S′ 2 FS ] and [S′ 1 ] ⌣ [S′ 2 ]=�S′ 1��S′ 2�[ω2p contradiction. Therefore, the support of S is connected. ⊓⊔
FS ],a
Example 1.23. Let f be a polynomial map of algebraic degree d ≥ 2onC k which
extends holomorphically to P k .IfB is a ball large enough centered at 0, then
f −1 (B) ⋐ B. Define Gn := d −n log + � f n �, where log + := max(log,0). As in
Theorem 1.16, we can show that Gn converge uniformly to a continuous p.s.h.
function G such that G ◦ f = dG. OnC k , the Green current T of f is equal to
dd c G and T p =(dd c G) p . The Green measure is equal to (dd c G) k .IfK denotes
the set of points in C k with bounded orbit, then μ is supported on K . Indeed,
outside K we have G = limd −n log� f n � and the convergence is locally uniform. It
follows that (dd c G) k = limd −kn (dd c log� f n �) k on C k \ K . One easily check that
(dd c log� f n �) k = 0 out of f −n (0). Therefore, (dd c G) k = 0onC k \ K .ThesetK
is called the filled Julia set. We can show that K is the zero set of G. In particular,
if f (z) =(z d 1 ,...,zd k ),thenG(z) =sup i log+ |zi|. One can check that the support of
T p is foliated (except for a set of zero measure with respect to the trace of T p )by
stable manifolds of dimension k − p and that μ = T k is the Lebesgue measure on
the torus {|zi| = 1, i = 1,...,k}.
Example 1.24. We consider Example 1.10.Letν be the Green measure of h on P 1 ,
i.e. ν = limd −n (h n ) ∗ (ωFS). Here, ωFS denotes also the Fubini-Study form on P 1 .