Discrete Holomorphic Local Dynamical Systems

Discrete Holomorphic Local Dynamical Systems 33
Remark 5.3. There are situations where one can use more or less directly the onedimensional
theory. For example, it is possible to study the so-called semi-direct
product of germs, namely germs f ∈ End(C 2 ,O) of the form
f (z1,z2)= � f1(z1), f2(z1,z2) � ,
or the so-called unfoldings, i.e., germs f ∈ End(Cn ,O) of the form
f (z1,...,zn)= � �
f1(z1,...,zn),z2,...,zn .
We refer to [J2] for the study of a particular class of semi-direct products, and to
[Ri1–2] for interesting results on unfoldings.
Let us begin assuming that the origin is a hyperbolic fixed point for an f ∈
End(C n ,O) not necessarily invertible. We then have a canonical splitting
C n = E s ⊕ E u ,
where E s (respectively, E u ) is the direct sum of the generalized eigenspaces associated
to the eigenvalues of dfO with modulus less (respectively, greater) than 1. Then
the first main result in this subject is the famous stable manifold theorem (originally
due to Perron [Pe] and Hadamard [H]; see [FHY, HK, HPS, Pes, Sh, AM] for proofs
in the C ∞ category, Wu [Wu] for a proof in the holomorphic category, and [A3] for
a proof in the non-invertible case):
Theorem 5.4 (Stable manifold theorem). Let f ∈ End(Cn ,O) be a holomorphic
local dynamical system with a hyperbolic fixed point at the origin. Then:
(i) the stable set Kf is an embedded complex submanifold of (a neighbourhood of
the origin in) Cn , tangent to Es at the origin;
(ii) there is an embedded complex submanifold Wf of (a neighbourhood of the origin
in) Cn , called the unstable set of f , tangent to Eu at the origin, such that
f |Wf is invertible, f −1 (Wf ) ⊆ Wf , and z ∈ Wf if and only if there is a sequence
{z−k}k∈N in the domain of f such that z0 = z and f (z−k)=z−k+1 for all k ≥ 1.
Furthermore, if f is invertible then Wf is the stable set of f −1 .
The proof is too involved to be summarized here; it suffices to say that both Kf
and Wf can be recovered, for instance, as fixed points of a suitable contracting operator
in an infinite dimensional space (see the references quoted above for details).
Remark 5.5. If the origin is an attracting fixed point, then E s = C n ,andKf is an open
neighbourhood of the origin, its basin of attraction. However, as we shall discuss
below, this does not imply that f is holomorphically linearizable, not even when
it is invertible. Conversely, if the origin is a repelling fixed point, then E u = C n ,
and Kf = {O}. Again, not all holomorphic local dynamical systems with a repelling
fixed point are holomorphically linearizable.
If a point in the domain U of a holomorphic local dynamical system with a hyperbolic
fixed point does not belong either to the stable set or to the unstable set,