Discrete Holomorphic Local Dynamical Systems

Uniformisation of Foliations by Curves 155
and meromorphic maps
πF : VF ��� X, ΠF : UF ��� X
such that, for any transversal T ⊂ X 0 , we have Q −1
F (T )=VT , qF |T = qT ,
πF | Q −1
F (T ) = πT ,etc.
Remark that if D ⊂ X 0 is a small disc contained in some leaf Lp of F , p ∈ D,
then Q −1
F (D) is naturally isomorphic to the product �Lp × D: foreveryq ∈ D, �Lq is
the same as �Lp, but with a different basepoint. More precisely, thinking to points
of �Lq as equivalence classes of paths starting at q, we see that for every q ∈ D
the isomorphism between �Lq and �Lp is completely canonical, once D is fixed and
because D is contractible. This means that D can be lifted, in a canonical way, to a
(D), transverse to the fibers. In this way, by varying D in
foliation by discs in Q −1
F
X 0 , we get in the full space VF a nonsingular foliation by curves � F , which projects
by QF to F 0 .
If γ : [0,1] → X 0 is a loop in a leaf, γ(0)=γ(1)=p, then this foliation � F permits
to define a monodromy map of the fiber �Lp into itself. This monodromy map is just
the covering transformation of �Lp corresponding to γ (which may be trivial, if the
holonomy of the foliation along γ is trivial).
In a similar way, in the space UF we get a canonically defined nonsingular foliation
by curves � F , which projects by PF to F 0 . And we have a fiberwise covering
FF : UF → VF
which is a local diffeomorphism, sending � F to � F .
γ = q
q F (X 0 )
γ from q to p
V F
L q
L p
γ = p
F
(over a leaf)
In the spaces UF and VF we also have the graphs of the sections pF and qF .
They are not invariant by the foliations � F and � F : in the notation above, with D
in a leaf and p,q ∈ D, the basepoint qF (q) ∈ �Lq corresponds to the constant path
γ(t) ≡ q, whereas the point of �Lq in the same leaf (of � F )ofqF (p) ∈ �Lp corresponds