Discrete Holomorphic Local Dynamical Systems

Discrete Holomorphic Local Dynamical Systems 37
Theorem 5.17 (Brjuno, 1971 [Brj2–3]). Let f ∈ End(C n ,O) be a holomorphic
local dynamical system such that f belongs to the Siegel domain, has no resonances,
and d fO is diagonalizable. Assume moreover that
+∞
∑
k=0
1
log
2k Then f is holomorphically linearizable.
1
Ω f (2 k+1 )
< +∞. (37)
Theorem 5.18. Let λ1,...,λn ∈ C be without resonances and such that
limsup
m→+∞
1
m log
1
Ω λ1,...,λn
(m) =+∞.
Then there exists f ∈ End(C n ,O), with d fO = diag(λ1,...,λn), not holomorphically
linearizable.
Remark 5.19. These theorems hold even without hyperbolicity assumptions.
Remark 5.20. It should be remarked that, contrarily to the one-dimensional case, it is
not yet known whether condition (37) is necessary for the holomorphic linearizability
of all holomorphic local dynamical systems with a given linear part belonging to
the Siegel domain. However, it is easy to check that if λ ∈ S 1 does not satisfy the
one-dimensional Brjuno condition then any f ∈ End(C n ,O) of the form
f (z)= � λ z1 + z 2 1 ,g(z)�
is not holomorphically linearizable: indeed, if ϕ ∈ End(C n ,O) is a holomorphic linearization
of f ,thenψ(ζ)=ϕ(ζ,O) is a holomorphic linearization of the quadratic
polynomial λ z + z 2 ,againstTheorem4.10.
Pöschel [Pö] shows how to modify (36) and(37) to get partial linearization results
along submanifolds, and Raissy [R1](seealso[Ro1]and[R2]) explains when
it is possible to pass from a partial linearization to a complete linearization even in
presence of resonances. Another kind of partial linearization results, namely “linearization
modulo an ideal”, can be found in [Sto]. Russmann [Rü] and Raissy [R4]
proved that in Theorem 5.17 one can replace the hypothesis “no resonances” by the
hypothesis “formally linearizable”, up to define Ω f (m) by taking the minimum only
over the non-resonant multiindeces. See also and Il’yachenko [I1] for an important
result related to Theorem 5.18. Raissy, in [R3], describes a completely different
way of proving the convergence of Poincaré-Dulac normal forms, based on torus
actions and allowing a detailed study of torsion phenomena. Finally, in [DG] results
in the spirit of Theorem 5.17 are discussed without assuming that the differential is
diagonalizable.