Discrete Holomorphic Local Dynamical Systems

26 Marco Abate
defined for |t| small enough. Hence,
1
sup
k k logαk < ∞,
as we wanted.
To estimate δk we have to take care of small divisors. First of all, for each k ≥ 2
we associate to δk a specific decomposition of the form
δk = ε −1
δk1 ···δkν k , (31)
with k > k1 ≥ ··· ≥ kν, k = k1 + ···+ kν and ν ≥ 2, and hence, by induction, a
specific decomposition of the form
δk = ε −1
l0 ε−1
l1
···ε −1
, (32)
lq
where l0 = k and k > l1 ≥ ··· ≥ lq ≥ 2. For m ≥ 2letNm(k) be the number of
factors ε −1
l in the expression (32)ofδk satisfying
εl < 1
4 Ωλ (m).
Notice that Ωλ (m) is non-increasing with respect to m and it tends to zero as m goes
to infinity. The next lemma contains the key estimate.
Lemma 4.14. For all m ≥ 2 we have
�
0, if k ≤ m ,
Nm(k) ≤ 2k
m − 1, if k > m.
Proof. We argue by induction on k. Ifl≤ k ≤ m we have εl ≥ Ωλ (m), and
hence Nm(k)=0.
Assume now k > m, sothat2k/m−1≥1. Write δk as in (31);wehaveafew
cases to consider.
Case 1: εk ≥ 1
4 Ωλ (m). Then
N(k)=N(k1)+···+ N(kν),
and applying the induction hypothesis to each term we get N(k) ≤ (2k/m) − 1.
Case 2: εk < 1 4 Ω λ (m). Then
N(k)=1 + N(k1)+···+ N(kν),
and there are three subcases.
Case 2.1: k1 ≤ m.Then
N(k)=1 ≤ 2k
− 1,
m
and we are done.
Case 2.2: k1 ≥ k2 > m. Then there is ν ′ such that 2 ≤ ν ′ ≤ ν and kν ′ > m ≥ kν ′ +1,
and we again have