Discrete Holomorphic Local Dynamical Systems

268 Tien-Cuong Dinh and Nessim Sibony
center a and of radius r, then the ratio between the volume of Y ∩Br and the volume
of a ball of radius r in C p decreases to mult(Y,a) when r decreases to 0.
Let τ : X1 → X2 be an open holomorphic map between complex manifolds of the
same dimension. Applying the above result to the graph of τ, we can show that for
any point a ∈ X1 and for a neighbourhoodU of a small enough, if z is a generic point
in X2 close enough to τ(a), the number of points in τ −1 (z)∩U does not depend on z.
We call this number the multiplicity or the local topological degree of τ at a.Wesay
that τ is a ramified covering of degree d if τ is open, proper and each fiber of τ contains
exactly d points counted with multiplicity. In this case, if Σ2 is the set of critical
values of τ and Σ1 := τ −1 (Σ2),thenτ : X1 \ Σ1 → X2 \ Σ2 is a covering of degree d.
We recall the notion of analytic space which generalizes complex manifolds and
their analytic subsets. An analytic space of dimension ≤ p is defined as a complex
manifold but a chart is replaced with an analytic subset of dimension ≤ p in an
open set of a complex Euclidean space. As in the case of analytic subsets, one can
decompose analytic spaces into irreducible components and into regular and singular
parts. The notions of dimension, of Zariski topology and of holomorphic maps
can be extended to analytic spaces. The precise definition uses the local ring of
holomorphic functions, see [GU,N]. An analytic space is normal if the local ring of
holomorphic functions at every point is integrally closed. This is equivalent to the
fact that for U open in Z holomorphic functions on reg(Z) ∩U which are bounded
near sing(Z) ∩U, are holomorphic on U. In particular, normal analytic spaces are
locally irreducible. A holomorphic map f : Z1 → Z2 between complex spaces is a
continuous map which induces morphisms from local rings of holomorphic functions
on Z2 to the ones on Z1. The notions of ramified covering, of multiplicity and
of open maps can be extended to normal analytic spaces. We have the following
useful result where �Z is called normalization of Z.
Theorem A.4. Let Z be an analytic space. Then there is a unique, up to a biholomorphic
map, normal analytic space �Z and a finite holomorphic map π : �Z → Z
such that
1. π −1 (reg(Z)) is a dense Zariski open set of �Z and π defines a bi-holomorphic
map between π −1 (reg(Z)) and reg(Z);
2. If τ : Z ′ → Z is a holomorphic map between analytic spaces, then there is a
unique holomorphic map h : Z ′ → �Z satisfying π ◦ h = τ.
In particular, holomorphic self-maps of Z can be lifted to holomorphic self-maps
of �Z.
Example A.5. Let π : C → C2 be the holomorphic map given by π(t)=(t2 ,t 3 ).This
map defines a normalization of the analytic curve {z3 1 = z22 } in C2 which is singular
at 0. The normalization of the analytic set {z1 = 0}∪{z3 1 = z22 } is the union of two
disjoint complex lines. The normalization of a complex curve (an analytic set of
pure dimension 1) is always smooth.