Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 259
Theorem 2.47. Let ( fs)s∈Σ be a family of polynomial-like maps as above. Assume
that fs0 has a large topological degree for some s0 ∈ Σ. ThenLk is Hölder continuous
in a neighbourhood of s0. In particular, the bifurcation currents B p are
moderate for 1 ≤ p ≤ dimΣ.
Let Λs denote the Perron-Frobenius operator associated to fs. For any Borel set
B, denote by ΩB the standard volume form on C k restricted to B. We first prove
some preliminary results.
Lemma 2.48. Let W be a neighbourhood of the filled Julia set Ks0 of fs0 . Then,
there is a neighbourhood Σ0 of s0 such that 〈μs,ϕ〉 depends continuously on (s,ϕ)
in Σ0 × PSH(W ).
Proof. We first replace Σ with a neighbourhood Σ0 of s0 small enough. So, for every
s ∈ Σ, the filled Julia set of fs is contained in U := f −1
s0 (V ) and in W. We also reduce
the size of V in order to assume that fs is polynomial-like on a neighbourhood of
U with values in a neighbourhood V ′ of V. Moreover, since 〈μs,ϕ〉 = 〈μs,Λs(ϕ)〉
and Λs(ϕ) depends continuously on (s,ϕ) in Σ × PSH(W ), we can replace ϕ with
Λ N s (ϕ) with N large enough and s ∈ Σ, in order to assume that W = V. Finally, since
Λs(ϕ) is defined on V ′ , it is enough to prove the continuity for ϕ p.s.h. on V such
that ϕ ≤ 1and〈ΩU,ϕ〉≥0. Denote by P the family of such functions ϕ.Sinceμs0
is PC, we have |〈μs0 ,ϕ〉| ≤ A for some constant A ≥ 1andforϕ∈P.LetP′ denote
the family of p.s.h. functions ψ such that ψ ≤ 2A and 〈μs0 ,ψ〉 = 0. The function
ϕ ′ := ϕ −〈μs0 ,ϕ〉 belongs to this family. Observe that P′ is bounded and therefore
if A ′ ≥ 1 is a fixed constant large enough, we have |〈ΩU,ψ〉| ≤ A ′ for ψ ∈ P ′ .
FixanintegerNlargeenough. By Theorem 2.33, Λ N s0 (ϕ′ ) ≤ 1/8 onV ′ and
|〈ΩU,Λ N s0 (ϕ′ )〉| ≤ 1/8 forϕ ′ as above. We deduce that 2Λ N s0 (ϕ′ ) −〈ΩU,2Λ N s0 (ϕ′ )〉
is a function in P, smaller than 1/2onV ′ . This function differs from 2Λ N s0 (ϕ) by a
constant. So, it is equal to 2Λ N s0 (ϕ) −〈ΩU,2Λ N s0 (ϕ)〉. WhenΣ0issmall enough, by
continuity, the operator Ls(ϕ) := 2Λ N s (ϕ)−〈ΩU ,2Λ N s (ϕ)〉 preserves P for s ∈ Σ0.
Therefore, since Λs preserves constant functions, we have
�
〈ΩU ,Λ N s (ϕ)〉 + 2 −1 Ls(ϕ) �
By induction, we obtain
Λ mN
s (ϕ) =Λ (m−1)N
s
= 〈ΩU,Λ N s (ϕ)〉 + 2 −1 Λ (m−1)N
s (Ls(ϕ)).
Λ mN
s (ϕ) =〈ΩU,Λ N s (ϕ)〉 + ···+ 2−m+1 〈ΩU,Λ N s (Lm−1 s
= � ΩU,Λ N� −m+1 m−1
s ϕ + ···+ 2 Ls (ϕ) �� + 2 −m L m s (ϕ)
(ϕ))〉 + 2 −m L m s (ϕ)
= � d −N
t ( f N s )∗ (ΩU ),ϕ + ···+ 2 −m+1 L m−1
s (ϕ) � + 2 −m L m s (ϕ).
We deduce from the above property of Ls that the last term converges uniformly to
0whenmgoes to infinity. The sum in the first term converges normally to the p.s.h.
function ∑m≥1 2−m+1Lm−1 s (ϕ), which depends continuously on (s,ϕ). Therefore,
Λ mN
s (ϕ) converge to a constant which depends continuously on (s,ϕ). But we know
that the limit is 〈μs,ϕ〉. The lemma follows. ⊓⊔