Discrete Holomorphic Local Dynamical Systems

282 Tien-Cuong Dinh and Nessim Sibony
The construction of U uses a kernel constructed in Bost-Gillet-Soulé[BGS]. We
call U the Green quasi-potential of S. Whenp = 1, two quasi-potentials of S differ
by a constant. So, the solution is unique if we require that 〈ωk FS ,U〉 = 0. In this
case, we have a bijective and bi-continuous correspondence S ↔ u between positive
closed (1,1)-currents S and their normalized quasi-potentials u.
By maximum principle, p.s.h. functions on a compact manifold are constant.
However, the interest of p.s.h. functions is their type of local singularities. S.T.
Yau introduced in [YA] the useful notion of quasi-p.s.h. functions. A quasi-p.s.h.
function is locally the difference of a p.s.h. function and a smooth one. Several properties
of quasi-p.s.h. functions can be deduced from properties of p.s.h. functions. If
u is a quasi-p.s.h. function on Pk there is a constant c > 0 such that ddcu ≥−cωFS.
So, ddcu is the difference of a positive closed (1,1)-current and a smooth positive
closed (1,1)-form: ddcu =(ddcu + cωFS) − cωFS. Conversely,ifSis a positive
closed (1,1)-current cohomologous to a real (1,1)-form α, there is a quasi-p.s.h.
function u, unique up to a constant, such that ddcu = S − α. The following proposition
is easily obtained using a convolution on the group of automorphisms of Pk ,
see Demailly [DEM] for analogous results on compact Kähler manifolds.
Proposition A.41. Let u be a quasi-p.s.h. function on P k such that dd c u ≥−ωFS.
Then, there is a sequence (un) of smooth quasi-p.s.h. functions decreasing to u such
that dd c un ≥−ωFS. In particular, if S is a positive closed (1,1)-current on P k ,then
there are smooth positive closed (1,1)-forms Sn converging to S.
A subset E of P k is pluripolar if it is contained in {u = −∞} where u is a quasip.s.h.
function. It is complete pluripolar if there is a quasi-p.s.h. function u such
that E = {u = −∞}. It is easy to check that analytic sets are complete pluripolar
and that a countable union of pluripolar sets is pluripolar. The following capacity is
close to a notion of capacity introduced by H. Alexander in [AL]. The interesting
point here is that our definition extends to general compact Kähler manifold [DS6].
We will see that the same idea allows to define the capacity of a current. Let P1
denote the set of quasi-p.s.h. functions u on P k such that max P k u = 0. The capacity
of a Borel set E in P k is
� �
cap(E) := inf exp
u∈P1
supu
E
.
The Borel set E is pluripolar if and only if cap(E) =0. It is not difficult to show
that when the volume of E tends to the volume of P k then cap(E) tends to 1.
The space of d.s.h. functions (differences of quasi-p.s.h. functions) and the complex
Sobolev space of functions on compact Kähler manifolds were introduced
by the authors in [DS6, DS11]. They satisfy strong compactness properties and
are invariant under the action of holomorphic maps. Using them as test functions,
permits to obtain several results in complex dynamics.
A function on P k is called d.s.h. if it is equal outside a pluripolar set to the
difference of two quasi-p.s.h. functions. We identify two d.s.h. functions if they are
equal outside a pluripolar set. Let DSH(P k ) denote the space of d.s.h. functions on
P k . We deduce easily from properties of p.s.h. functions that DSH(P k ) is contained