Discrete Holomorphic Local Dynamical Systems

142 Marco Brunella
where, as usual, �v(q)�Poin is the norm of v(q) measured with the Poincaré metric
on Lq. Recall that this “metric” is identically zero when Lq is parabolic, so that F is
equal to −∞ on the intersection of U 0 with parabolic leaves.
Proposition 6.2. The function F above is either plurisubharmonic or
identically −∞.
Proof. Let T ⊂ U 0 be a transversal to F 0 ,andletUT be the corresponding covering
tube. Put on the fibers of UT their Poincaré metric. The vector field v induces a
nonsingular vertical vector field on UT along pT (T ), which we denote again by v.
Due to the arbitrariness of T , and by a connectivity argument, we need just to verify
that the function on T defined by
F(z)=log�v(pT (z)�Poin
is either plurisubharmonic or identically −∞. That is, the fiberwise Poincarémetric
on UT has a plurisubharmonic variation.
The upper semicontinuity of F being evident (see e.g. [Suz, §3] or [Kiz]), let us
consider the submean inequality over discs in T .
Take a closed disc S ⊂ T as in Theorem 5.1, i.e. satisfying hypotheses (a) and (b)
of Section 5. By that Theorem, and by choosing an increasing sequence of compact
subsets Kj in ∂US, we can find a sequence of relatively compact domains Ω j ⊂ US,
j ∈ N, such that:
(i) the relative boundary of Ω j in US is a real analytic Levi-flat hypersurface Mj ⊂
US, with boundary Γj ⊂ ∂US, filledbyaS 1 -family of graphs of holomorphic
sections of US with boundary values in Γj;
(ii) for every z ∈ S,thefiberΩj(z)=Ω j ∩P −1
S (z) is a disc, centered at pS(z);more-
over, for z ∈ ∂S we have ∪ +∞
j=1 Ω j(z)=P −1
S (z).
Note that one cannot hope that the exhaustive property in (ii) holds also for z in the
interior of S.
We may apply to Ω j, whose boundary is Levi-flat and hence pseudoconvex, the
result of Yamaguchi discussed in Section 2, more precisely Proposition 2.2. Itsays
that the function on S
Fj(z)=log�v(pS(z)� Poin( j),
where �v(pS(z)�Poin( j) is the norm with respect to the Poincaré metriconthedisc
Ω j(z), is plurisubharmonic. Hence we have at the center 0 of S � D the submean
inequality:
Fj(0) ≤ 1
� 2π
Fj(e
2π 0
iθ )dθ.
We now pass to the limit j → +∞. Foreveryz ∈ ∂S we have Fj(z) → F(z),
by the exhaustive property in (ii) above. Moreover, we may assume that Ω j(z) is
an increasing sequence for every z ∈ ∂S (and in fact for every z ∈ S, but this is not
important), so that Fj(z) converges to F(z) in a decreasing way, by the monotonicity