Discrete Holomorphic Local Dynamical Systems

Uniformisation of Foliations by Curves 149
Remark that the space Pω(V,U) is not empty, for it contains at least all the
constant nonpositive functions on V.
Suppose that Pω(V,U) satisfies the following condition:
(A) the functions in Pω(V,U) are locally uniformly bounded from above: for every
z ∈ V there exists a neighbourhood Vz ⊂ V of z and a constant cz such that
ϕ|Vz ≤ cz for every ϕ ∈ Pω(V,U).
Then we can introduce the upper envelope
Φ(z)= sup
ϕ∈Pω (V,U)
ϕ(z) ∀z ∈ V
and its upper semicontinuous regularization
The function
Φ ∗ (z)=limsupΦ(w)
∀z ∈ V.
w→z
Φ ∗ : V → [0,+∞)
is identically zero on U, upper semicontinuous, and ω-plurisubharmonic (Brelot-
Cartan [Kli]), hence it belongs to the space Pω(V,U). Moreover, by results of Bedford
and Taylor [BeT, Kli] the wedge product (dd c Φ ∗ + ω) n is well defined, as a
locally finite measure on V, and it is identically zero outside U:
(dd c Φ ∗ + ω) n ≡ 0 onV \U.
Indeed, let B ⊂ V \U be a ball around which ω has a potential. Let Pω(B,Φ ∗ )
be the space of ω-plurisubharmonic functions ψ on B such that limsup z→w ψ(z) ≤
Φ ∗ (w) for every w ∈ ∂B. LetΨ ∗ be the regularized upper envelope of the family
Pω(B,Φ ∗ ) (which is bounded from above by the maximum principle). Remark that
Φ ∗ |B belongs to Pω(B,Φ ∗ ),andsoΨ ∗ ≥ Φ ∗ on B.By[BeT], Ψ ∗ satisfies the homogeneous
Monge-Ampère equation (dd c Ψ ∗ + ω) n = 0onB, with Dirichlet boundary
condition limsup z→wΨ ∗ (z)=Φ ∗ (w), w ∈ ∂B (“balayage”). Then the function �Φ ∗
on V, which is equal to Ψ ∗ on B and equal to Φ ∗ on V \ B, still belongs to Pω(V,U),
and it is everywhere not smaller than Φ ∗ . Hence, by definition of Φ ∗ ,wemusthave
�Φ ∗ = Φ ∗ ,i.e.Φ ∗ = Ψ ∗ on B and so Φ ∗ satisfies the homogeneous Monge-Ampère
equation on B.
Suppose now that the following condition is also satisfied:
(B) Φ ∗ : V → [0,+∞) is exhaustive on V \ U: for every c > 0, the subset
{Φ ∗ < c}\U is relatively compact in V \U.
Roughly speaking, this means that the function Φ ∗ solves on V \ U the homogeneous
Monge-Ampère equation, with boundary conditions 0 on ∂U and +∞ on the
“boundary at infinity” of V \U.