Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 221
Let b := logN(logδ0) −1 .Wehave
for N large. We also have
cNe −δ b 0 = cNe −N ≤ e −N/2
e−λ b + eλ b λ
=
2 ∑
n≥0
2nb2n (2n)! ≤ eλ 2b2 .
Therefore, if λ := uεb −2 with a fixed u > 0 small enough
−λ Nε
2e
�
e−λ b �N + eλ b
2
≤ 2e −ε2 b −2 (1−u)Nu = 2e −2N(logN) −2 hε
for some constant hε > 0. We deduce from the previous estimates that
ν � SN(ψ ′ ) ≥ Nε � ≤ e −N(logN)−2 hε
for N large. A similar estimate holds for −ψ ′ .So,ψ ′ satisfies the weak LDT. ⊓⊔
We deduce from Theorem 1.94, Corollaries 1.41 and 1.42 the following result
[DNS].
Theorem 1.102. Let f be a holomorphic endomorphism of Pk of algebraic degree
d ≥ 2. Then the equilibrium measure μ of f satisfies the weak large deviations theorem
for bounded d.s.h. observables and also for Hölder continuous observables.
More precisely, if a function ψ is bounded d.s.h. or Hölder continuous, then for
every ε > 0 there is a constant hε > 0 such that
μ
�
z ∈ P k :
for all N large enough.
�
�
� 1
�
�N
N−1
∑
n=0
ψ ◦ f n � �
�
�
(z) −〈μ,ψ〉 � > ε
�
≤ e −N(logN)−2 hε
The exponential estimate on Λ n (ψ) is crucial in the proofs of the previous
results. It is nearly an estimate in sup-norm. Note that if �Λ n (ψ)� L ∞ (μ) converge
exponentially fast to 0 then ψ satisfies the LDT. This is the case for Hölder continuous
observables in dimension 1, following a result by Drasin-Okuyama [DO],
and when f is a generic map in higher dimension, see Remark 1.71. TheLDTwas
recently obtained in dimension 1 by Xia-Fu in [X] for Lipschitz observables.
Exercise 1.103. Let g : X → X be a continuous map on a compact metric space X.
Deduce from Birkhoff’s theorem that any ergodic invariant measure of g is a limit of
for an appropriate x.
N−1 1
N
∑
n=0
δ g n (x)