Discrete Holomorphic Local Dynamical Systems

200 Tien-Cuong Dinh and Nessim Sibony
1.5 Equidistribution of Varieties
In this paragraph, we consider the inverse images by f n of varieties in P k .The
geometrical method in the last paragraph is quite difficult to apply here. Indeed,
the inverse image of a generic variety of codimension p < k is irreducible of degree
O(d pn ). The pluripotential method that we introduce here is probably the right
method for the equidistribution problem. Moreover, it should give some precise
estimates on the convergence, see Remark 1.71.
The following result, due to the authors, gives a satisfactory solution in the case
of hypersurfaces. It was proved for Zariski generic maps by Fornæss-Sibony in
[FS3, S3] and for maps in dimension 2 by Favre-Jonsson in [FJ]. More precise
results are given in [DS9] andin[FJ, FJ1] whenk = 2. The proof requires some
self-intersection estimates for currents, due to Demailly-Méo.
Theorem 1.64. Let f be an endomorphism of algebraic degree d ≥ 2 of P k .Let
Em denote the union of the totally invariant proper analytic sets in P k which are
minimal, i.e. do not contain smaller ones. Let S be a positive closed (1,1)-current
of mass 1 on P k whose local potentials are not identically −∞ on any component of
Em. Then, d −n ( f n ) ∗ (S) converge weakly to the Green (1,1)-current T of f .
The following corollary gives a solution to the equidistribution problem for
hypersurfaces: the exceptional hypersurfaces belong to a proper analytic set in the
parameter space of hypersurfaces of a given degree.
Corollary 1.65. Let f , T and Em be as above. If H is a hypersurface of degree s in
P k , which does not contain any component of Em, thens −1 d −n ( f n ) ∗ [H] converge to
T in the sense of currents.
Note that ( f n ) ∗ [H] is the current of integration on f −n (H) where the components
of f −n (H) are counted with multiplicity.
Sketch of the proof of Theorem 1.64. We can write S = T + dd c u where u is a
p.s.h. function modulo T , that is, the difference of quasi-potentials of S and of T .
Subtracting from u a constant allows to assume that 〈μ,u〉 = 0. We call u the dynamical
quasi-potential of S. SinceT has continuous quasi-potentials, u satisfies
analogous properties that quasi-p.s.h. functions do. We are mostly concerned with
the singularities of u.
The total invariance of T and μ implies that the dynamical quasi-potential
of d −n ( f n ) ∗ (S) is equal to un := d −n u ◦ f n . We have to show that this sequence
of functions converges to 0 in L 1 (P k ).Sinceu is bounded from above, we have
limsupun ≤ 0. Assume that un do not converge to 0. By Hartogs’ lemma, see
Proposition A.20,thereisaballB and a constant λ > 0 such that un ≤−λ on B for
infinitely many indices n. It follows that u ≤−λ d n on f n (B) for such an index n.On
the other hand, the exponential estimate in Theorem A.22 implies that �e α|u| � L 1 ≤ A
for some positive constants α and A independent of u. If the multiplicity of f at