Discrete Holomorphic Local Dynamical Systems

Uniformisation of Foliations by Curves 143
property of the Poincaré metric. It follows that the boundary integral in the submean
inequality above converges, as j → +∞, to 1
� 2π
2π 0 F(eiθ )dθ (which may be −∞, of
course).
Concerning Fj(0), it is sufficient to observe that, obviously, F(0) ≤ Fj(0), because
Ω j(0) ⊂ P −1
S (0),andsoF(0) ≤ liminf j→+∞ Fj(0). In fact, and because Ω j(0)
U S
S
v
Mj+1 Ωj+1 Mj Ωj
Γ j+1
Γ j
p S (S)
is increasing, Fj(0) converges to some value c in [−∞,+∞), but we may have the
strict inequality F(0) < c if Ω j(0) do not exhaust P −1
S (0). Therefore the above submean
inequality gives, at the limit,
F(0) ≤ 1
2π
� 2π
0
F(e iθ )dθ
that is, the submean inequality for F on S.
Take now an arbitrary closed disc S ⊂ T, centered at some point p ∈ T. By
Lemma 6.1 and the remarks before it, we may approximate S by a sequence of
closed discs S j with the same center p and satisfying moreover hypotheses (a) and
(b) before Theorem 5.1 (unless R = T , but in that case UT = T × C and F ≡−∞).
More precisely, if ϕ : D → T is a parametrization of S, ϕ(0)=p, then we may uniformly
approximate ϕ by a sequence of embeddings ϕ j : D → T , ϕ j(0)=p, such
that S j = ϕ j(D) satisfies the assumptions of Theorem 5.1. Hence we have, by the
previous arguments and for every j,
F(p) ≤ 1
2π
� 2π
0
F(ϕ j(e iθ ))dθ
and passing to the limit, using Fatou Lemma, and taking into account the upper
semicontinuity of F, we finally obtain
� 2π
1
F(p) ≤ limsup F(ϕ j(e
j→+∞ 2π 0
iθ ))dθ ≤ 1
2π
≤ 1
� 2π
F(ϕ(e
2π
iθ ))dθ.
0
� 2π
0
limsupF(ϕ
j(e
j→+∞
iθ ))dθ ≤