Discrete Holomorphic Local Dynamical Systems

Discrete Holomorphic Local Dynamical Systems 47
Now, a combinatorial argument (inspired by Camacho and Sad [CS]; see also
[Ca] and[T]) shows that if we have f ∈ End(C 2 ,O) tangent to the identity with
an isolated fixed point, and such that applying the reduction of singularities to the
lifted map ˜f starting from a singular direction [v] ∈ P 1 (C) =E weenduponly
with (⋆2) singularities, then the index of ˜f at [v] along E must be a non-negative
rational number. But the first Chern class of NE is −1; so there must be at least one
singular directions whose index is not a non-negative rational number. Therefore the
reduction of singularities must yield at least one dicritical or (⋆1) singularity, and
hence a Fatou flower for our map f , completing the proof of Theorem 6.17.
Actually, we have proved the following slightly more precise result:
Theorem 6.31 (Abate, 2001 [A2]). Let E be a (not necessarily compact) Riemann
surface inside a 2-dimensional complex manifold M, and take f ∈ End(M,E)
tangential to E. Let p ∈ E be a singular point of f such that ιp( f ,E) /∈ Q + .
Then there exist a Fatou flower for f at p. In particular, if fo ∈ End(C2 ,O) is
a non-dicritical holomorphic local dynamical system tangent to the identity with
an isolated fixed point at the origin, and [v] ∈ P1 � � (C) is a singular direction such
that ι [v] f ,P1 (C) /∈ Q + , where f is the lift of fo to the blow-up of the origin, then
fo has a Fatou flower tangent to [v].
Remark 6.32. This latter statement has been generalized in two ways. Degli
Innocenti [D1] has proved that we can allow E to be singular at p (but irreducible;
in the reducible case one has to impose conditions on the indeces of f along all
irreducible components of E passing through p). Molino [Mo], on the other hand,
has proved that the statement still holds assuming only ιp( f ,E) �= 0, at least for f
of order 2 (and E smooth at p); it is natural to conjecture that this should be true for
f of any order.
As already remarked, the reduction of singularities via blow-ups seem to work only
in dimension 2. This leaves open the problem of the validity of something like
Theorem 6.17 in dimension n ≥ 3; see [AT1]and[Ro2] for some partial results.
As far as I know, it is widely open, even in dimension 2, the problem of describing
the stable set of a holomorphic local dynamical system tangent to the identity, as
well as the more general problem of the topological classification of such dynamical
systems. Some results in the case of a dicritical singularity are presented in [BM];
for non-dicritical singularities a promising approach in dimension 2 is described
in [AT3].
Let f ∈ End(M,E), whereE is a smooth Riemann surface in a 2-dimensional
complex manifold M, and assume that f is tangential; let E o denote the complement
in E of the singular points of f . The connection ∇ on NEo described above induces
a connection (still denoted by ∇) onN ⊗ν f
Eo .
Definition 6.33. In this setting, a geodesic is a curve σ : I → Eo such that
∇ σ ′X −1
f (σ ′ ) ≡ O.