Discrete Holomorphic Local Dynamical Systems

Discrete Holomorphic Local Dynamical Systems 15
So setting ϕ = ˜ϕ ◦ ψ, we have defined a function ϕ with the required properties
on P + j . To extend it to the whole basin B it suffices to put
ϕ(z)=ϕ � f k (z) � − k, (13)
where k ∈ N is the first integer such that f k (z) ∈ P + j . ⊓⊔
A way to construct the conjugation ϕ as limit of hyperbolic linearizations given
by Theorem 2.4 is described in [U3].
Remark 3.7. It is possible to construct petals that cannot be contained in any sector
strictly smaller than Σ j. TodosoweneedanF-invariant subset ˆHε of C∗ \ R− containing ˜Hε and containing eventually every half-line issuing from the origin
(but R− ). For M >> 1andC > 0 large enough, replace the straight lines bounding
˜Hε on the left of Rew = −M by the curves
�
C log|Rew| if r = 1 ,
|Imw| =
C|Rew| 1−1/r if r > 1.
Then it is not too difficult to check that the domain ˆHε so obtained is as desired (see
[CG]).
So we have a complete description of the dynamics in the neighbourhood of
the origin. Actually, Camacho has pushed this argument even further, obtaining a
complete topological classification of one-dimensional holomorphic local dynamical
systems tangent to the identity (see also [BH, Theorem 1.7]):
Theorem 3.8 (Camacho, 1978 [C]; Shcherbakov, 1982 [S]). Let f ∈ End(C,0) be
a holomorphic local dynamical system tangent to the identity with multiplicity r + 1
at the fixed point. Then f is topologically locally conjugated to the map
g(z)=z − z r+1 .
Remark 3.9. Camacho’s proof ([C]; see also [Br2, J1]) shows that the topological
conjugation can be taken smooth in a punctured neighbourhood of the origin.
Jenkins [J1] also proved that if f ∈ End(C,0) is tangent to the identity with
multiplicity 2 and the topological conjugation is actually real-analitic in a punctured
neighbourhood of the origin, with real-analytic inverse, then f is locally holomorphically
conjugated to z−z 2 . Finally, Martinet and Ramis [MR]haveprovedthatifa
germ f ∈ End(C,0) tangent to the identity is C 1 -conjugated (in a full neighbourhood
of the origin) to g(z)=z?z r+1 , then the conjugation can be chosen holomorphic or
antiholomorphic.
The formal classification is simple too, though different (see, e.g., Milnor [Mi]):
Proposition 3.10. Let f ∈ End(C,0) be a holomorphic local dynamical system tangent
to the identity with multiplicity r + 1 at the fixed point. Then f is formally
conjugated to the map
g(z)=z − z r+1 + β z 2r+1 , (14)