Discrete Holomorphic Local Dynamical Systems

18 Marco Abate
a lower half-plane. Moreover, we have H ± j (z + 1) =H± j (z)+1; therefore using
the projection π(z) =exp(2πiz) we can induce holomorphic maps h ± j : π(V ± j ) →
π(W ± j ),whereπ(V + j ) and π(W + j ) are pointed neighbourhoods of the origin, and
π(V − j ) and π(W − j ) are pointed neighbourhoods of ∞ ∈ P1 (C).
It is possible to show that setting h + j (0) =0 one obtains a holomorphic germ
h + j ∈ End(C,0), and that setting h− j (∞)=∞ one obtains a holomorphic germ h+ j ∈
End � P1 (C),∞ � . Furthermore, denoting by λ + j (respectively, λ − j ) the multiplier of
h + j at 0 (respectively, of h− j at ∞), it turns out that
r
∏(λ j=1
+ j λ − j )=exp�4π 2 Resit( f ) � . (19)
Now, if we replace f by a holomorphic local conjugate ˜f = ψ−1 ◦ f ◦ ψ, and
denote by ˜h ± j the corresponding germs, it is not difficult to check that (up to a cyclic
renumbering of the petals) there are constants α j, β j ∈ C∗ such that
˜h − j (z)=α jh − � �
z
j
and ˜h + j (z)=α j+1h + � �
z
j . (20)
β j
This suggests the introduction of an equivalence relation on the set of 2r-uple of
germs (h ± 1 ,...,h± r ).
Definition 3.12. Let Mr denote the set of 2r-uple of germs h =(h ± 1 ,...,h± r ), with
h + j ∈ End(C,0), h− j ∈ End� P 1 (C),∞ � , and whose multipliers satisfy (19). We shall
say that h, ˜h ∈ Mr are equivalent if up to a cyclic permutation of the indeces we have
(20) for suitable α j, β j ∈ C ∗ . We denote by Mr the set of all equivalence classes.
The procedure described above allows then to associate to any f ∈ End(C,0) tangent
to the identity with multiplicity r + 1anelementμ f ∈ Mr.
Definition 3.13. Let f ∈ End(C,0) be tangent to the identity. The element μ f ∈ Mr
given by this procedure is the sectorial invariant of f .
Then the holomorphic classification proved by Écalle and Voronin is
Theorem 3.14 (Écalle, 1981 [É2–3]; Voronin, 1981 [Vo]). Let f , g ∈ End(C,0) be
two holomorphic local dynamical systems tangent to the identity. Then f and g are
holomorphically locally conjugated if and only if they have the same multiplicity,
the same index and the same sectorial invariant. Furthermore, for any r ≥ 1, β ∈ C
and μ ∈ Mr there exists f ∈ End(C,0) tangent to the identity with multiplicity r +1,
index β and sectorial invariant μ.
Remark 3.15. In particular, holomorphic local dynamical systems tangent to the
identity give examples of local dynamical systems that are topologically conjugated
without being neither holomorphically nor formally conjugated, and of local
dynamical systems that are formally conjugated without being holomorphically
conjugated. See also [Na, Tr].
β j