Discrete Holomorphic Local Dynamical Systems

254 Tien-Cuong Dinh and Nessim Sibony
is Lipschitz with respect to dist L 1 (K) . Indeed, if ϕ,ψ are in L 1 (K), wehaveforthe
standard volume form Ω on Ck �
�Λ(ϕ) − Λ(ψ)� L1 (K) =
K
|Λ(ϕ − ψ)|Ω ≤ d −1
�
t
f −1 |ϕ − ψ| f
(K)
∗ (Ω).
since f −1 (K) ⊂ K and f ∗ (Ω) is bounded on f −1 (K), this implies that
�Λ(ϕ) − Λ(ψ)� L 1 (K) ≤ const�ϕ − ψ� L 1 (K) .
Since P is compact, the functions in P are uniformly bounded above on U.
Therefore, replacing P by the family of Λ(ϕ) with ϕ ∈ P allows to assume that
functions in P are uniformly bounded above on V. On the other hand, since μ is
PC, μ is bounded on P. Without loss of generality, we can assume that P is the set
of functions ϕ such that 〈μ,ϕ〉≥0andϕ ≤ 1. In particular, P is invariant under Λ.
Let D be the family of d.s.h. functions ϕ − Λ(ϕ) with ϕ ∈ P. This is a compact
subset of DSH(V) which is invariant under Λ, and we have 〈μ,ϕ ′ 〉 = 0forϕ ′ in D.
Consider a function ϕ ∈ P. Observe that �ϕ := ϕ −〈μ,ϕ〉 is also in P. Define
�Λ := λ −1 Λ with λ the constant in Theorem 2.33. We deduce from that theorem
that �Λ(�ϕ) is in P. Moreover,
�Λ � ϕ − Λ(ϕ) � = � Λ � �ϕ − Λ(�ϕ) � = � Λ(�ϕ) − Λ � � Λ(�ϕ) � .
Therefore, D is invariant under � Λ. This is the key point in the proof. Observe that
we can extend distL1 (K) to DSH(V ) and that �Λ is Lipschitz with respect to this
pseudo-distance.
Let ν be a smooth probability measure with support in K. We have seen that
d−n t ( f n ) ∗ (ν) converge to μ.Ifϕ is d.s.h. on V, then
〈d −n
t ( f n ) ∗ (ν),ϕ〉 = 〈ν,Λ n (ϕ)〉.
Define for ϕ in P, ϕ ′ := ϕ − Λ(ϕ).Wehave
〈μ,ϕ〉 = lim
n→∞ 〈ν,Λ n (ϕ)〉
= 〈ν,ϕ〉−∑ 〈ν,Λ
n≥0
n (ϕ)〉−〈ν,Λ n+1 (ϕ)〉
= 〈ν,ϕ〉−∑ λ
n≥0
n 〈ν, �Λ n (ϕ ′ )〉.
Since ν is smooth with support in K, it is Lipschitz with respect to dist L 1 (K) .We
deduce from Lemma 1.19 which is also valid for a pseudo-distance, that the last
series defines a Hölder continuous function on D. WeuseheretheinvarianceofD
under � Λ. Finally, since the map ϕ ↦→ ϕ ′ is Lipschitz on P, we conclude that μ is
Hölder continuous on P with respect to dist L 1 (K) . �