Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 289
Exercise A.55. Let μ be a non-zero positive measure which is WPC. Define
�u� ∗ μ := |〈μ,u〉| + min�Θ�1/2
with Θ as above. Show that �·� ∗ μ defines a norm which is equivalent to �·�W ∗.
Exercise A.56. Show that the capacity of R is positive if and only if R is PB.
Exercise A.57. Let S be a positive closed (p, p)-current of mass 1 with positive
Lelong number at a point a.LetH be a hyperplane containing a such that S and [H]
are wedgeable. Show that the Lelong number of S ∧ [H] at a isthesameifweconsider
it as a current on P k or on H. IfR is a positive closed current of bidimension
(p − 1, p − 1) on H, show that US(R) ≤ U S∧[H](R)+c where c > 0 is a constant independent
of S,R and H. Deduce that PB currents have no positive Lelong numbers.
Exercise A.58. Let K be a compact subset in C k ⊂ P k . Let S1,...,Sp be
positive closed (1,1)-currents on P k . Assume that their quasi-potentials are bounded
on P k \ K. Show that S1,...,Sp are wedgeable. Show that the wedge-product
S1 ∧ ...∧ Sp is continuous for Hartogs’ convergence.
Exercise A.59. Let S and S ′ be positive closed (p, p)-currents on P k such that
S ′ ≤ S. Assume that S isPB(resp.PC).ShowthatS ′ is PB (resp. PC).
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