Discrete Holomorphic Local Dynamical Systems

172 Tien-Cuong Dinh and Nessim Sibony
For a holomorphic map f on P k , a point a in P k is critical if f is not injective in a
neighbourhood of a or equivalently the multiplicity of f at a in the fiber f −1 ( f (a)) is
strictly larger than 1, see Theorem A.3. We say that a is a critical point of multiplicity
m if the multiplicity of f at a in the fiber f −1 ( f (a)) is equal to m + 1.
Proposition 1.6. Let f be a holomorphic endomorphism of algebraic degree d ≥ 2
of P k . Then, the critical set of f is an algebraic hypersurface of degree (k+1)(d −1)
counted with multiplicity.
Proof. If F is a lift of f to C k+1 , the Jacobian Jac(F) is a homogeneous polynomial
of degree (k + 1)(d − 1). The zero set of Jac(F) in P k is exactly the critical set of f .
The result follows. ⊓⊔
Let C denote the critical set of f . The orbit C , f (C ), f 2 (C ),... is either a hypersurface
or a countable union of hypersurfaces. We say that f is postcritically finite
if this orbit is a hypersurface, i.e. has only finitely many irreducible components.
Besides very simple examples, postcritically finite maps are difficult to construct,
because the image of a variety is often a variety of larger degree; so we have to
increase the multiplicity in order to get only finitely many irreducible components.
We give few examples of postcritically finite maps, see [FS, FS7].
Example 1.7. We can check that for d ≥ 2and(1 − 2λ ) d = 1
f [z0 : ···: zk] :=[z d 0 : λ (z0 − 2z1) d : ···: λ (z0 − 2zk) d ]
is postcritically finite. For some parameters α ∈ C and 0 ≤ l ≤ d,themap
fα[z] :=[z d 0 : zd1 : zd2 + αzd−l
1 z l 2 ]
is also postcritically finite. In particular, for f0[z] =[zd 0 : zd 1 : zd 2 ], the associated
critical set is equal to {z0z1z2 = 0} which is invariant under f0. So, f0 is postcritically
finite.
Arguing as above, using Bézout’s theorem, we can prove that if Y is an analytic
set of pure codimension p in P k then f −1 (Y ) is an analytic set of pure
codimension p. Its degree, counting with multiplicity, is equal to d p deg(Y). Recall
that the degree deg(Y ) of Y is the number of points in the intersection of Y
with a generic projective subspace of dimension p. We deduce that the pull-back
operator f ∗ on the Hodge cohomology group H p,p (P k ,C) is simply a multiplication
by d p .Since f is a ramified covering of degree d k , f∗ ◦ f ∗ is the multiplication by
d k . Therefore, the push-forward operator f∗ acting on H p,p (P k ,C) is the multiplication
by d k−p . In particular, the image f (Y ) of Y by f is an analytic set of pure
codimension p and of degree d k−p deg(Y ), counted with multiplicity.
We now introduce the Fatou and Julia sets associated to an endomorphism. The
following definition is analogous to the one variable case.
Definition 1.8. The Fatou set of f is the largest open set F in P k where the sequence
of iterates ( f n )n≥1 is locally equicontinuous. The complement J of F is called the
Julia set of f .