Discrete Holomorphic Local Dynamical Systems

36 Marco Abate
as I know, the problem of finding canonical formal normal forms when f belongs to
the Siegel domain is still open (see [J2] for some partial results).
It should be remarked that, in the hyperbolic case, the problem of formal linearization
is equivalent to the problem of smooth linearization. This has been proved
by Sternberg [St1–2] and Chaperon [Ch]:
Theorem 5.14 (Sternberg, 1957 [St1–2]; Chaperon, 1986 [Ch]). Assume we have
f,g∈ End(C n ,O) two holomorphic local dynamical systems, with f locally invertible
and with a hyperbolic fixed point at the origin. Then f and g are formally
conjugated if and only if they are smoothly locally conjugated. In particular, f is
smoothly linearizable if and only if it is formally linearizable. Thus if there are no
resonances then f is smoothly linearizable.
Even without resonances, the holomorphic linearizability is not guaranteed. The
easiest positive result is due to Poincaré[Po] who, using majorant series, proved the
following
Theorem 5.15 (Poincaré, 1893 [Po]). Let f ∈ End(C n ,O) be a locally invertible
holomorphic local dynamical system in the Poincaré domain. Then f is holomorphically
linearizable if and only if it is formally linearizable. In particular, if there
are no resonances then f is holomorphically linearizable.
Reich [Re2] describes holomorphic normal forms when dfO belongs to the Poincaré
domain and there are resonances (see also [ÉV]); Pérez-Marco [P8] discusses the
problem of holomorphic linearization in the presence of resonances (see also Raissy
[R1]).
When dfO belongs to the Siegel domain, even without resonances, the formal
linearization might diverge. To describe the known results, let us introduce the following
definition:
Definition 5.16. For λ1,...,λn ∈ C and m ≥ 2set
Ωλ1,...,λn (m)=min� |λ k1 kn
1 ···λn −λ j| � � k1,...,kn ∈ N, 2≤k1+···+kn ≤ m, 1≤ j≤n � .
If λ1,...,λn are the eigenvalues of dfO, we shall write Ω f (m) for Ω λ1,...,λn (m).
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It is clear that Ω f (m) �= 0forallm ≥ 2 if and only if there are no resonances. It
is also not difficult to prove that if f belongs to the Siegel domain then
lim
m→+∞ Ω f (m)=0,
which is the reason why, even without resonances, the formal linearization might be
diverging, exactly as in the one-dimensional case. As far as I know, the best positive
and negative results in this setting are due to Brjuno [Brj2–3] (see also [Rü] and
[R4]), and are a natural generalization of their one-dimensional counterparts, whose
proofs are obtained adapting the proofs of Theorems 4.13 and 4.6 respectively: