Discrete Holomorphic Local Dynamical Systems

288 Tien-Cuong Dinh and Nessim Sibony
respect to the trace measure of Si+1 ∧ ...∧ Sp. Then, this condition is symmetric on
S1,...,Sp. The wedge-product S1 ∧...∧Sp is a positive closed (p, p)-current of mass
�S1�...�Sp� supported on supp(S1) ∩ ...∩ supp(Sp). It depends linearly on each
variable and is symmetric on the variables. If S (n)
i
sense, then the S (n)
i are wedgeable and S (n)
1
converge to Si in the Hartogs’
∧ ...∧ S(n)
p converge to S1 ∧ ...∧ Sp.
We discuss now currents with Hölder continuous super-potential and moderate
currents. The space Ck−p+1(P k ) admits natural distances distα, with α > 0, defined
by
distα(R,R ′ ) := sup
�Φ�C α ≤1
|〈R − R ′ ,Φ〉|,
where Φ is a smooth (p − 1, p − 1)-form on P k . The norm C α on Φ is the sum of
the C α -norms of its coefficients for a fixed atlas of P k . The topology associated to
distα coincides with the weak topology. Using the theory of interpolation between
Banach spaces [T1], we obtain for β > α > 0that
dist β ≤ distα ≤ c α,β [dist β ] α/β
where c α,β > 0 is a constant. So, a function on Ck−p+1(P k ) is Hölder continuous
with respect to distα if and only if it is Hölder continuous with respect to dist β .The
following proposition is useful in dynamics.
Proposition A.51. The wedge-product of positive closed currents on P k with Hölder
continuous super-potentials has a Hölder continuous super-potential. Let S be a
positive closed (p, p)-current with a Hölder continuous super-potential. Then, the
Hausdorff dimension of S is strictly larger than 2(k − p). Moreover, S is moderate,
i.e. for any bounded family F of d.s.h. functions on P k , there are constants c > 0
and α > 0 such that �
e α|u| dσS ≤ c
for every u in F ,whereσS is the trace measure of S.
Exercise A.52. Show that there is a constant c > 0suchthat
Hint: use the compactness of P1 in L 1 .
cap(E) ≥ exp(−c/volume(E)).
Exercise A.53. Let (un) be a sequence of d.s.h. functions such that ∑�un�DSH
is finite. Show that ∑un converge pointwise out of a pluripolar set to a d.s.h.
function. Hint: write un = u + n − u− n with u± n ≤ 0, �u± n �DSH � �un�DSH and
dd c u ± n ≥−�un�DSHωFS.
Exercise A.54. If χ is a convex increasing function on R with bounded derivative
and u is a d.s.h. function, show that χ ◦u is d.s.h. If χ is Lipschitz and u is in W ∗ (P k ),
show that χ ◦ u is in W ∗ (P k ). Prove that bounded d.s.h. functions are in W ∗ (P k ).
Show that DSH(P k ) and W ∗ (P k ) are stable under the max and min operations.