Discrete Holomorphic Local Dynamical Systems

170 Tien-Cuong Dinh and Nessim Sibony
Zelevinsky shows that Md(P k ) \ Hd(P k ) is an irreducible algebraic variety [GK].
We will be concerned in this section mostly with holomorphic maps. We show that
they are open and their topological degree, i.e. the number of points in a generic
fiber, is equal to d k . We recall here the classical Bézout’s theorem which is a central
tool for the dynamics in P k .
Theorem 1.1 (Bézout). Let P1,...,Pk be homogeneous polynomials in C k+1 of degrees
d1,...,dk respectively. Let Z denote the set of common zeros of Pi, inP k ,i.e.
the set of points [z] such that Pi(z)=0 for 1 ≤ i ≤ k. If Z is discrete, then the number
of points in Z, counted with multiplicity, is d1 ...dk.
The multiplicity of a point a in Z can be defined in several ways. For instance,
if U is a small neighbourhood of a and if P ′
i are generic homogeneous polynomials
of degrees di close enough to Pi, then the hypersurfaces {P ′
i = 0} in Pk intersect
transversally. The number of points of the intersection in U does not depend on the
choice of P ′
i and is the multiplicity of a in Z.
Proposition 1.2. Let f be an endomorphism of algebraic degree d of P k . Then for
every a in P k , the fiber f −1 (a) contains exactly d k points, counted with multiplicity.
In particular, f is open and defines a ramified covering of degree d k .
Proof. For the multiplicity of f and the notion of ramified covering, we refer to Appendix
A.1. Let f =[F0 : ···: Fk] be an expression of f in homogeneous coordinates.
Consider a point a =[a0 : ··· : ak] in P k . Without loss of generality, we can assume
a0 = 1, hence a =[1:a1 : ··· : ak]. The points in f −1 (a) are the common zeros, in
P k , of the polynomials Fi − aiF0 for i = 1,...,k.
We have to check that the common zero set is discrete, then Bézout’s theorem
asserts that the cardinality of this set is equal to the product of the degrees of Fi −
aiF0, i.e.tod k . If the set were not discrete, then the common zero set of Fi − aiF0
in C k+1 is analytic of dimension ≥ 2. This implies that the set of common zeros of
the Fi’s, 0 ≤ i ≤ k, inC k+1 is of positive dimension. This is impossible when f is
holomorphic. So, f is a ramified covering of degree d k . In particular, it is open.
Note that when f is a map in Md(P k )\Hd(P k ) with indeterminacy set I, we can
prove that the generic fibers of f : P k \I → P k contains at most d k −1 points. Indeed,
for every a, the hypersurfaces {Fi − aiF0 = 0} in P k contain I. ⊓⊔
Periodic points of order n, i.e. points which satisfy f n (z)=z, play an important
role in dynamics. Here, f n := f ◦···◦ f , n times, is the iterate of order n of f .
Periodic points of order n of f are fixed points of f n which is an endomorphism
of algebraic degree d n . In the present case, their counting is simple. We have the
following result.
Proposition 1.3. Let f be an endomorphism of algebraic degree d ≥ 2 in P k .Then
the number of fixed points of f , counted with multiplicity, is equal to (d k+1 − 1)/
(d − 1). In particular, the number of periodic points of order n of f is d kn + o(d kn ).