Discrete Holomorphic Local Dynamical Systems

260 Tien-Cuong Dinh and Nessim Sibony
Using the same approach as in Theorem 2.34, we prove the following result.
Theorem 2.49. Let fs, s0 and W be as in Theorem 2.47 and Lemma 2.48.LetKbe
a compact subset of W such that f −1
s0 (K) is contained in the interior of K. There
is a neighbourhood Σ0 of s0 such that if P is a bounded family of p.s.h. functions
on W , then (s,ϕ) ↦→ 〈μs,ϕ〉 is Hölder continuous on Σ0 × P with respect to the
pseudo-distance distL1 (K) on P.
Proof. We replace Σ with Σ0 as in Lemma 2.48. It is not difficult to check that
(s,ϕ) ↦→ (s,Λs(ϕ)) is locally Lipschitz with respect to dist L 1 (K) . So, replacing (s,ϕ)
by (s,Λ N s (ϕ)) with N large enough allows to assume that W = V. Let � P be the
set of (s,ϕ) in Σ × PSH(V ) such that ϕ ≤ 1and〈μs,ϕ〉 ≥0. By Lemma 2.48,
such functions ϕ belong to a compact subset of PSH(V ). It is enough to prove that
(s,ϕ) ↦→〈μs,ϕ〉 is Hölder continuous on � P.
Let � D denote the set of (s,ϕ − Λs(ϕ)) with (s,ϕ) ∈ � P. Consider the operator
�Λ(s,ψ) :=(s,λ −1 Λs(ψ)) on D as in Theorem 2.34 where λ < 1 is a fixed constant
close enough to 1. Theorem 2.33 and the continuity in Lemma 2.48 imply that �Λ
preserves � D. Therefore, we only have to follow the arguments in Theorem 2.34. ⊓⊔
Proof of Theorem 2.47. We replace Σ with a small neighbourhood of s0. Observe
that logJac( fs), s ∈ Σ, is a bounded family of p.s.h. functions on U. By Theorem
2.49, it is enough to show that s ↦→ logJac( fs) is Hölder continuous with respect to
dist L 1 (K) .
We also deduce from Theorem A.22 that 〈ΩK,e λ |logJac( fs)| 〉≤A for some positive
constants λ and A. Reducing V and Σ allows to assume that Jac(F), their derivatives
and the vanishing order of Jac(F) are bounded on Σ ×U by some constant m.
Fix a constant α > 0 small enough and a constant A > 0 large enough. Define
ψ(s) := 〈ΩK,logJac( fs)〉. Consider s and t in Σ such that r := �s−t� is smaller than
a fixed small constant. We will compare |ψ(s)−ψ(t)| with r λα in order to show that
ψ is Hölder continuous with exponent λα.DefineS := {z ∈ U, Jac( fs) < 2r 2α }.
We will bound separately
and
〈Ω K\S,logJac( fs) − logJac( ft)〉
〈ΩK∩S,logJac( fs) − logJac( ft)〉.
Note that ψ(s) − ψ(t) is the sum of the above two integrals.
Consider now the integral on K \ S. The following estimates are only valid on
K \ S. Since the derivatives of Jac(F) is bounded, we have Jac( ft) ≥ r 2α . It follows
that the derivatives on t of logJac( ft) is bounded by Ar −2α . We deduce that
Therefore,
|logJac( fs) − logJac( ft)|≤Ar 1−2α .
|〈Ω K\S,logJac( fs) − logJac( ft)〉| ≤ �Ω K\S�Ar 1−2α ≤ r λα .