Discrete Holomorphic Local Dynamical Systems

216 Tien-Cuong Dinh and Nessim Sibony
Theorem 1.94. Let g : (X,F ,ν) → (X,F ,ν) be a map with bounded Jacobian as
above. Define Fn := g −n (F ).Letψ be a bounded real-valued measurable function.
Assume there are constants δ > 1 and c > 0 such that
� ν,e δ n |E(ψ|Fn)−〈ν,ψ〉| � ≤ c for every n ≥ 0.
Then ψ satisfies a weak large deviations theorem. More precisely, for every ε > 0,
there exists a constant hε > 0 such that
� �
�
� 1
ν x ∈ X : � ψ ◦ g
� N
n � �
�
�
(x) −〈ν,ψ〉 � > ε ≤ e
� −N(logN)−2 hε
for all N large enough 6 .
N−1
∑
n=0
We first prove some preliminary lemmas. The following one is a version of the
classical Bennett’s inequality see [DZ, Lemma 2.4.1].
Lemma 1.95. Let ψ be an observable such that �ψ� L ∞ (ν) ≤ b for some constant
b ≥ 0, and E(ψ)=0.Then
for every λ ≥ 0.
E(e λψ ) ≤ e−λ b + eλ b
2
Proof. We can assume λ = 1. Consider first the case where there is a measurable
set A such that ν(A)=1/2. Let ψ0 be the function which is equal to −b on A and
to b on X \ A. Wehaveψ 2 0 = b2 ≥ ψ 2 .Sinceν(A)=1/2, we have E(ψ0)=0. Let
g(t)=a0t 2 + a1t + a2, be the unique quadratic function such that h(t) := g(t) − e t
satisfies h(b)=0andh(−b)=h ′ (−b)=0. We have g(ψ0)=e ψ0.
Since h ′′ (t)=2a0 − e t admits at most one zero, h ′ admits at most two zeros. The
fact that h(−b) =h(b) =0 implies that h ′ vanishes in ] − b,b[. Hence h ′ admits
exactly one zero at −b and another one in ] − b,b[. We deduce that h ′′ admits a
zero. This implies that a0 > 0. Moreover, h vanishes only at −b, b and h ′ (b) �=0. It
follows that h(t) ≥ 0on[−b,b] because h is negative near +∞. Thus, e t ≤ g(t) on
[−b,b] and then e ψ ≤ g(ψ).
Since a0 > 0, if an observable φ satisfies E(φ)=0, then E(g(φ)) is an increasing
function of E(φ 2 ). Now, using the properties of ψ and ψ0, we obtain
E(e ψ ) ≤ E(g(ψ)) ≤ E(g(ψ0)) = E(e ψ0
e
)= −b + eb .
2
This completes the proof under the assumption that ν(A)=1/2 for some measurable
set A.
The general case is deduced from the previous particular case. Indeed, it is
enough to apply the first case to the disjoint union of (X,F ,ν) with a copy
6 In the LDT for independent random variables, there is no factor (logN) −2 in the estimate.