Discrete Holomorphic Local Dynamical Systems

202 Tien-Cuong Dinh and Nessim Sibony
we deduce that ui = 0 on supp(μ).DefineS1 ∨ S2 := T + dd c max(u1,u2). It is easy
to check that S1 ∨ S2 is an element of C .AnelementS is said to be minimal if
S = S1 ∨ S2 implies S1 = S2 = S. It is clear that T is not minimal if C contains other
currents. In fact, for S in C ,wehaveT ∨S = T . A current of integration on a totally
invariant hypersurface is a minimal element. It is likely that C is generated by a
finite number of currents, the operation ∨, convex hulls and limits.
Example 1.68. If f is the map given in Example 1.11, the exceptional set Em is the
union of the k + 1 attractive fixed points
[0:···:0:1:0:···:0].
The convergence of s −1 d −n ( f n ) ∗ [H] towards T holds for hypersurfaces H of degree
s which do not contain these points. If π : C k+1 \{0} →P k is the canonical projection,
the Green (1,1)-current T of f is given by π ∗ (T )=dd c (maxi log|zi|), or
equivalently T = ωFS + dd c v where
1
v[z0 : ···: zk] := max log|zi|−
0≤i≤k 2 log(|z0| 2 + ···+ |zk| 2 ).
The currents Ti of integration on {zi = 0} belong to C and Ti = T + dd c ui with
ui := log|zi|−max j log|z j|. These currents are minimal. If α0, ..., αk are positive
real numbers such that α := 1−∑αi is positive, then S := αT +∑αiTi is an element
of C .WehaveS = T + dd c u with u := ∑αiui. The current S is minimal if and only
if α = 0. One can obtain other elements of C using the operator ∨. We show that C
is infinite dimensional. Define for A :=(α0,...,αk) with 0 ≤ αi ≤ 1and∑αi = 1
the p.s.h. function vA by
vA := ∑αi log|zi|.
If A is a family of such (k + 1)-tuples A,define
vA := sup vA.
A∈A
Then, we can define a positive closed (1,1)-currents SA on P k by π ∗ (SA )=dd c vA .
It is clear that SA belongs to C and hence C is of infinite dimension.
The equidistribution problem in higher codimension is much more delicate and
is still open for general maps. We first recall the following lemma.
Lemma 1.69. For every δ > 1, there is a Zariski dense open set H ∗
d (Pk ) in Hd(Pk )
and a constant A > 0 such that for f in H ∗
d (Pk ), the maximal multiplicity δn of f n
at a point in Pk is at most equal to Aδ n . In particular, the exceptional set of such a
map f is empty when δ < d.
Proof. Let X be a component of a totally invariant analytic set E of pure dimension
p ≤ k − 1. Then, f permutes the components of E. We deduce that X is totally invariant
under f n for some n ≥ 1. Lemma 1.48 implies that the maximal multiplicity