Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 169
1.1 Basic Properties and Examples
Let f : P k → P k be a holomorphic endomorphism. Such a map is always induced
by a polynomial self-map F =(F0,...,Fk) on C k+1 such that F −1 (0)={0} and the
components Fi are homogeneous polynomials of the same degree d ≥ 1. Given an
endomorphism f , the associated map F is unique up to a multiplicative constant and
is called a lift of f to C k+1 . From now on, assume that f is non-invertible, i.e. the
algebraic degree d is at least 2. Dynamics of an invertible map is simple to study.
If π : C k+1 \{0}→P k is the natural projection, we have f ◦ π = π ◦ F. Therefore,
dynamics of holomorphic maps on P k can be deduced from the polynomial case in
C k+1 . We will count preimages of points, periodic points, introduce Fatou and Julia
sets and give some examples.
It is easy to construct examples of holomorphic maps in P k . The family of
homogeneous polynomial maps F of a given degree d is parametrized by a complex
vector space of dimension Nk,d := (k + 1)(d + k)!/(d!k!). The maps satisfying
F −1 (0)={0} define a Zariski dense open set. Therefore, the parameter space
Hd(P k ), of holomorphic endomorphisms of algebraic degree d, is a Zariski dense
open set in P N k,d−1 , in particular, it is connected.
If f : C k → C k is a polynomial map, we can extend f to P k but the extension is
not always holomorphic. The extension is holomorphic when the dominant homogeneous
part f + of f , satisfies ( f + ) −1 (0)={0}. Here, if d is the maximal degree in
the polynomial expression of f ,then f + is composed by the monomials of degree
d in the components of f . So, it is easy to construct examples using products of one
dimensional polynomials or their pertubations.
A general meromorphic map f : P k → P k of algebraic degree d is given in homogeneous
coordinates by
f [z0 : ···: zk]=[F0 : ···: Fk],
where the components Fi are homogeneous polynomials of degree d without common
factor, except constants. The map F :=(F0,...,Fk) on C k+1 is still called a lift
of f . In general, f is not defined on the analytic set I = {[z] ∈ P k ,F(z)=0} which is
of codimension ≥ 2 since the Fi’s have no common factor. This is the indeterminacy
set of f which is empty when f is holomorphic.
It is easy to check that if f is in Hd(P k ) and g is in H d ′(P k ), the composition f ◦g
belongs to H dd ′(P k ). This is in general false for meromorphic maps: the algebraic
degree of the composition is not necessarily equal to the product of the algebraic
degrees. It is enough to consider the meromorphic involution of algebraic degree k
�
1
f [z0 : ···: zk] := : ···:
z0
1
� �
z0 ...zk
=
zk z0
: ···: z0
�
...zk
.
zk
The composition f ◦ f is the identity map.
We say that f is dominant if f (P k \ I) contains a non-empty open set. The space
of dominant meromorphic maps of algebraic degree d, is denoted by Md(P k ).It
is also a Zariski dense open set in P N k,d−1 . A result by Guelfand, Kapranov and