Discrete Holomorphic Local Dynamical Systems

12 Marco Abate
When |w| is so large that ψ −1 (w) belongs to the domain of definition of f ,the
composition F = ψ ◦ f ◦ ψ −1 makes sense, and we have
F(w)=w + 1 + O(w −1/r ). (6)
Thus to study the dynamics of f in a neighbourhood of the origin in Σ j it suffices to
study the dynamics of F in a neighbourhood of infinity.
The first observation is that when Rew is large enough then
ReF(w) > Rew + 1
2 ;
this implies that for δ small enough H δ is F-invariant (and thus P j,δ is f -invariant).
Furthermore, by induction one has
ReF k (w) > Rew + k
2
for all w ∈ H δ , which implies that F k (w) → ∞ in H δ (and f k (z) → 0inP j,δ )as
k → ∞.
Now we claim that the argument of wk = F k (w) tends to zero. Indeed, (6)and(7)
yield
wk
k
= w
k
k−1 1
+ 1 +
k
∑
l=0
O(w −1/r
l ) ;
hence Cesaro’s theorem on the averages of a converging sequence implies
wk
k
(7)
→ 1, (8)
and so argwk → 0ask→ ∞. Going back to Pj,δ , this implies that f k (z)/| f k (z)|→v + j
for every z ∈ Pj,δ . Since furthermore Pj,δ is centered about v + j , every orbit converging
to 0 tangent to v + j must intersect Pj,δ , and thus we have proved that Pj,δ is an
attracting petal.
Arguing in the same way with f −1 we get repelling petals; unfortunately, the
petals obtained so far are too small to form a full pointed neighbourhood of the origin.
In fact, as remarked before each Pj,δ is contained in a sector centered about v + j
of amplitude π/r; therefore the repelling and attracting petals obtained in this way
do not intersect but are tangent to each other. We need larger petals.
So our aim is to find an f -invariant subset P + j of Σ j containing Pj,δ and which
is tangent at the origin to a sector centered about v + j of amplitude strictly greater
than π/r. To do so, first of all remark that there are R, C > 0suchthat
|F(w) − w − 1|≤ C
|w| 1/r
as soon as |w| > R. Choose ε ∈ (0,1) and select δ > 0sothat4rδ < R −1 and
ε > 2C(4rδ) 1/r .Then|w| > 1/(4rδ) implies
|F(w) − w − 1| < ε/2.
(9)