Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 275
each variable. Let u1,...,un be p.s.h. functions on X. Then χ(u1,...,un) is p.s.h. In
particular, the function max(u1,...,un) is p.s.h.
We call complete pluripolar set the pole set {u = −∞} of a p.s.h. function and
pluripolar set a subset of a complete pluripolar one. Pluripolar sets are of Hausdorff
dimension ≤ 2k − 2, in particular, they have zero Lebesgue measure. Finite and
countable unions of (locally) pluripolar sets are (locally) pluripolar. In particular,
finite and countable unions of analytic subsets are locally pluripolar.
Proposition A.19. Let E be a closed pluripolar set in X and u a p.s.h. function on
X \ E, locally bounded above near E. Then the extension of u to X given by
is a p.s.h. function.
u(z) := limsupu(w)
w→z
w∈X\E
for z ∈ E,
The following result describes compactness properties of p.s.h. functions, see
[HO].
Proposition A.20. Let (un) be a sequence of p.s.h. functions on X, locally bounded
from above. Then either it converges locally uniformly to −∞ on a component of
X or there is a subsequence (uni ) which converges in Lp
loc (X) to a p.s.h. function
u for every p with 1 ≤ p < ∞. In the second case, we have limsupuni ≤ u with
equality outside a pluripolar set. Moreover, if K is a compact subset of X and if h is a
continuous function on K such that u < honK,thenuni < h on K for i large enough.
The last assertion is the classical Hartogs’ lemma. It suggests the following notion
of convergence introduced in [DS10]. Let (un) be a sequence of p.s.h. functions
converging to a p.s.h. function u in L 1 loc (X). We say that the sequence (un) converges
in the Hartogs’ sense or is H-convergent if for any compact subset K of X there
are constants cn converging to 0 such that un + cn ≥ u on K. In this case, Hartogs’
lemma implies that un converge pointwise to u. If(un) decreases to a function u,
not identically −∞, thenu is p.s.h. and (un) converges in the Hartogs’ sense. The
following result is useful in the calculus with p.s.h. functions.
Proposition A.21. Let u be a p.s.h. function on an open subset D of C k .LetD ′ ⋐ D
be an open set. Then, there is a sequence of smooth p.s.h. functions un on D ′ which
decreases to u.
The functions un can be obtained as the standard convolution of u with some
radial function ρn on C k . The submean inequality for u allows to choose ρn so that
un decrease to u.
The following result, see [HO2], may be considered as the strongest compactness
property for p.s.h. functions. The proof can be reduced to the one dimensional case
by slicing.