Discrete Holomorphic Local Dynamical Systems

182 Tien-Cuong Dinh and Nessim Sibony
Exercise 1.31. Let f be an endomorphism of algebraic degree d ≥ 2ofP k .LetX
be an analytic set of pure dimension p in an open set U ⊂ P k . Show that for every
compact K ⊂ U
1
limsup
n→∞ n logvolume( f n (X ∩ K)) ≤ plogd.
Hint. For an appropriate cut-off function χ, estimate �
X χ( f n ) ∗ (ω p
FS ).
1.3 Other Constructions of the Green Currents
In this paragraph, we give other methods, introduced and developped by the authors,
in order to construct the Green currents and Green measures for meromorphic maps.
We obtain estimates on the Perron-Frobenius operator and on the thickness of the
Green measure, that will be applied in the stochastic study of the dynamical system.
A key point here is the use of d.s.h. functions as observables.
We first present a recent direct construction of Green (p, p)-currents using superpotentials
1 . Super-potentials are a tool in order to compute with positive closed
(p, p)-currents. They play the same role as potentials for bidegree (1,1) currents.
In dynamics, they are used in particular in the equidistribution problem for algebraic
sets of arbitrary dimension and allow to get some estimates on the speed of
convergence.
Theorem 1.32. Let S be a positive closed (p, p)-current of mass 1 on P k . Assume
that the super-potential of S is bounded. Then d −pn ( f n ) ∗ (S) converge to the Green
(p, p)-current T p of f . Moreover, T p has a Hölder continuous super-potential.
Sketch of proof. We refer to Appendix A.2 and A.4 for an introduction to superpotentials
and to the action of holomorphic maps on positive closed currents. Recall
that f ∗ and f∗ act on H p,p (Pk ,C) as the multiplications by d p and dk−p respectively.
So, if S is a positive closed (p, p)-current of mass 1, then � f ∗ (S)� = d p and
� f∗(S)� = dk−p since the mass can be computed cohomologically. Let Λ denote the
operator d−p+1 f∗ acting on Ck−p+1(Pk ), the convex set of positive closed currents
of bidegree (k − p + 1,k − p + 1) and of mass 1. It is continuous and it takes values
also in Ck−p+1(Pk ).LetV , U , Un denote the super-potentials of d−p f ∗ (ω p
), S
FS
and d−pn ( f n ) ∗ (S) respectively. Consider a quasi-potential U of mean 0 of S which
is a DSH current satisfying ddcU = S − ω p
FS . The following computations are valid
for S smooth and can be extended to all currents S using a suitable regularization
procedure.
By Theorem A.35 in the Appendix, the current d−p f ∗ (U) is DSH and satisfies
dd c� d −p f ∗ (U) � = d −p f ∗ (S) − d −p f ∗ (ω p
FS ).
1 These super-potentials correspond to super-potentials of mean 0 in [DS10].