Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 177
The following lemma, due to Dinh-Sibony [DS4, DS10], implies the above
proposition and can be applied in a more general setting. Here, we apply it to
Λ := f m with m large enough and to the above smooth function v. We choose
α := 1, A := �Df m �∞ and d is replaced with d m .
Lemma 1.19. Let K be a metric space with finite diameter and Λ : K → Kbea
Lipschitz map: �Λ(a) − Λ(b)� ≤A�a − b� with A > 0. Here,�a − b� denotes the
distance between two points a, b in K. Let v be an α-Hölder continuous function on
K with 0 < α ≤ 1. Then, ∑n≥0 d −n v ◦Λ n converges pointwise to a function which is
β -Hölder continuous on K for every β such that 0 < β < α and β ≤ logd/logA.
Proof. By hypothesis, there is a constant A ′ >0suchthat|v(a) − v(b)|≤A ′ �a − b�α .
Define A ′′ := �v�∞. SinceKhas finite diameter, A ′′ is finite and we only have to
consider the case where �a − b�≪1. If N is an integer, we have
�
�
�
�
� ∑ d
n≥0
−n v ◦ Λ n (a) − ∑ d
n≥0
−n v ◦ Λ n �
�
�
(b) �
�
≤ ∑ d
0≤n≤N
−n |v ◦ Λ n (a) − v ◦ Λ n (b)| + ∑ d
n>N
−n |v ◦ Λ n (a) − v ◦ Λ n (b)|
≤ A ′ ∑ d
0≤n≤N
−n �Λ n (a) − Λ n (b)� α + 2A ′′ ∑ d
n>N
−n
� �a − b� α ∑
0≤n≤N
d −n A nα + d −N .
If A α ≤ d, the last sum is of order at most equal to N�a − b� α + d −N .Foragiven
0 < β < α, choose N �−β log�a−b�/logd. So, the last expression is � �a−b� β .
In this case, the function is β -Hölder continuous for every 0 < β < α. When
A α > d, thesumis� d −N A Nα �a − b� α + d −N .ForN �−log�a − b�/logA, the
last expression is � �a − b� β with β := logd/logA. Therefore, the function is
β -Hölder continuous. ⊓⊔
Remark 1.20. Lemma 1.19 still holds for K with infinite diameter if v is Hölder
continuous and bounded. We can also replace the distance on K with any positive
symmetric function on K × K which vanishes on the diagonal. Consider a
family ( fs) of endomorphisms of P k depending holomorphically on s in a space
of parameters Σ. In the above construction of the Green current, we can locally
on Σ, choose vs(z) smooth such that dd c s,zvs(z) ≥−ωFS(z). Lemma 1.19 implies
that the Green function gs(z) of fs is locally Hölder continuous on (s,z) in Σ × P k .
Then, ωFS(z)+dd c s,z gs(z) is a positive closed (1,1)-current on Σ × P k . Its slices by
{s}×P k are the Green currents Ts of fs.
We want to use the properties of the Green currents in order to establish some
properties of the Fatou and Julia sets. We will show that the Julia set coincides
with the Julia set of order 1. We recall the notion of Kobayashi hyperbolicity on
a complex manifold M. Letp be a point in M and ξ a tangent vector of M at
p. Consider the holomorphic maps τ : Δ → M on the unit disc Δ in C such that