Discrete Holomorphic Local Dynamical Systems

20 Marco Abate
Definition 3.19. The number n is the parabolic multiplicity of f ,andα ∈ C is the
index of f ;theiterative residue of f is then given by
Resit( f )=
nq + 1
− α.
2
Proposition 3.20 (Camacho). Let f ∈ End(C,0) be a holomorphic local dynamical
system with multiplier λ ∈ S 1 , and assume that λ is a primitive root of the unity
of order q. Assume that f q �≡ id, and has parabolic multiplicity n ≥ 1. Then f is
topologically conjugated to
g(z)=λ z − z nq+1 .
Theorem 3.21 (Leau-Fatou). Let f ∈ End(C,0) be a holomorphic local dynamical
system with multiplier λ ∈ S 1 , and assume that λ is a primitive root of the unity
of order q. Assume that f q �≡ id, and let n ≥ 1 be the parabolic multiplicity of f .
Then f q has multiplicity nq+1, and f acts on the attracting (respectively, repelling)
petals of f q as a permutation composed by n disjoint cycles. Finally, Kf = Kf q.
Furthermore, it is possible to define the sectorial invariant of such a holomorphic
local dynamical system, composed by 2nq germs whose multipliers still satisfy (19),
and the analogue of Theorem 3.14 holds.
4 One Complex Variable: The Elliptic Case
We are left with the elliptic case:
f (z)=e 2πiθ z + a2z 2 + ···∈C0{z}, (21)
with θ /∈ Q. It turns out that the local dynamics depends mostly on numerical properties
of θ. The main question here is whether such a local dynamical system is
holomorphically conjugated to its linear part. Let us introduce a bit of terminology.
Definition 4.1. We shall say that a holomorphic dynamical system of the form (21)
is holomorphically linearizable if it is holomorphically locally conjugated to its linear
part, the irrational rotation z ↦→ e2πiθ z. In this case, we shall say that 0 is a Siegel
point for f ; otherwise, we shall say that it is a Cremer point.
It turns out that for a full measure subset B of θ ∈ [0,1]\Q all holomorphic local
dynamical systems of the form (21) are holomorphically linearizable. Conversely,
the complement [0,1] \ B is a Gδ -dense set, and for all θ ∈ [0,1] \ B the quadratic
polynomial z ↦→ z2 +e2πiθ z is not holomorphically linearizable. This is the gist of the
results due to Cremer, Siegel, Brjuno and Yoccoz we shall describe in this section.
The first worthwhile observation in this setting is that it is possible to give a topological
characterization of holomorphically linearizable local dynamical systems.
Definition 4.2. We shall say that p is stable for f ∈ End(M, p) if it belongs to the
interior of Kf .