Discrete Holomorphic Local Dynamical Systems

Discrete Holomorphic Local Dynamical Systems 35
See also Ruggiero [Ru] for similar results for semi-superattracting (one eigenvalue
zero, one eigenvalue different from zero) germs in C 2 .
Coming back to hyperbolic dynamical systems, the holomorphic and even the
formal classification are not as simple as the topological one. The main problem is
caused by resonances.
Definition 5.9. Let f ∈ End(C n ,O) be a holomorphic local dynamical system, and
let denote by λ1,...,λn ∈ C the eigenvalues of dfO. Aresonance for f is a relation
of the form
λ k1 kn
1 ···λn − λ j = 0 (35)
for some 1 ≤ j ≤ n and some k1,...,kn ∈ N with k1 +···+kn ≥ 2. When n = 1there
is a resonance if and only if the multiplier is a root of unity, or zero; but if n > 1
resonances may occur in the hyperbolic case too.
Resonances are the obstruction to formal linearization. Indeed, a computation completely
analogous to the one yielding Proposition 4.4 shows that the coefficients of a
formal linearization have in the denominators quantities of the form λ k1 kn
1 ···λn −λj;
hence
Proposition 5.10. Let f ∈ End(C n ,O) be a locally invertible holomorphic local dynamical
system with a hyperbolic fixed point and no resonances. Then f is formally
conjugated to its differential d fO.
In presence of resonances, even the formal classification is not that easy. Let us
assume, for simplicity, that dfO is in Jordan form, that is
with ε1,...,εn−1 ∈{0,1}.
P1(z)=(λ1z,ε2z1 + λ2z2,...,εnzn−1 + λnzn),
Definition 5.11. We shall say that a monomial z k1
1 ···zkn
n in the j-th coordinate of f
is resonant if k1 + ···+ kn ≥ 2andλ k1 kn
1 ···λn = λ j.
Then Proposition 5.10 can be generalized to (see [Ar, p. 194] or [IY, p. 53] for a
proof):
Proposition 5.12 (Poincaré, 1893 [Po]; Dulac, 1904 [Du]). Let f ∈ End(C n ,O)
be a locally invertible holomorphic local dynamical system with a hyperbolic fixed
point. Then it is formally conjugated to a g ∈ C0[[z1,...,zn]] n such that dgO is in
Jordan normal form, and g has only resonant monomials.
Definition 5.13. The formal series g is called a Poincaré-Dulac normal form of f .
The problem with Poincaré-Dulac normal forms is that they are not unique. In
particular, one may wonder whether it could be possible to have such a normal
form including finitely many resonant monomials only (as happened, for instance,
in Proposition 3.10).
This is indeed the case (see, e.g., Reich [Re1]) when f belongs to the Poincaré
domain, that is when dfO is invertible and O is either attracting or repelling. As far