Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 215
The ASIP implies other stochastic results, see [PS], in particular, the law of the
iterated logarithm. With our notations, it implies that for ϕ as above
limsup
N→∞
SN(ϕ)
σ � N loglog(Nσ 2 )
= 1 μ-almost everywhere.
Dupont’s approach is based on the Philipp-Stout’s result applied to a Bernoulli
system and a quantitative Bernoulli property of the equilibrium measure of f ,i.e.a
construction of a coding tree. We refer to Dupont and Przytycki-Urbanski-Zdunik
[PU] for these results.
Recall the Bernoulli property of μ which was proved by Briend in [BJ1]. The
dimension one case is due to Mañé [MA] and Heicklen-Hoffman [HH]. Denote
by (Σ,ν,σ) the one-sided dk-shift, whereΣ := {1,...,dk } N , ν is the probability
measure on Σ induced by the equilibrium probability measure on {1,...,dk } and
σ : Σ → Σ is the dk to 1 map defined by σ(α0,α1,...)=(α1,α2,...).
Theorem 1.92. Let f and μ be as above. Then (Pk , μ, f ) is measurably conjugated
to (Σ,ν,σ). More precisely, there is a measurable map π : Σ → Pk , defined out of a
set of zero ν-measure, which is invertible out of a set of zero μ-measure, such that
π∗(ν)=μ and f = π ◦ σ ◦ π−1 μ-almost everywhere.
The proof uses a criterion, the so called tree very weak Bernoulli property (treevwB
for short) due to Hoffman-Rudolph [HR]. One can use Proposition 1.51 in
order to check this criterion.
The last stochastic property we consider here is the large deviations theorem. As
above, we first recall the classical result in probability theory.
Theorem 1.93. Let Z1,Z2,... be independent random variables on (X,F ,ν), identically
distributed with values in R, and of mean zero, i.e. E(Z1)=0. Assume also
that for t ∈ R, exp(tZn) is integrable. Then, the limit
I(ε) := − lim
N→∞ logν
�� � �
����
Z1 + ···+
�
ZN �
� > ε .
N �
exists and I(ε) > 0 for ε > 0.
The theorem estimates the size of the set where the average is away from zero,
the expected value. We have
�� � �
����
Z1 + ···+
�
ZN �
ν
� > ε ∼ e
N � −NI(ε) .
Our goal is to give an analogue for the equilibrium measure of endomorphisms
of P k . We first prove an abstract result corresponding to the above Gordin’s result
for the central limit theorem.
Consider a dynamical system g : (X,F ,ν) → (X,F ,ν) as above where ν is an
invariant probability measure. So, g ∗ defines a linear operator of norm 1 from L 2 (ν)
into itself. We say that g has bounded Jacobian if there is a constant κ > 0 such that
ν(g(A)) ≤ κν(A) for every A ∈ F . The following result was obtained in [DNS].