Bernoulli coding map and almost sure invariance principle for endomorphisms of $${\mathbb P^k}$$

Abstract

Let f be an holomorphic endomorphism of P k and µ be its measure of maximal entropy. We prove an almost sure invariance principle for the systems (P k , f, µ). Our class U of observables includes the Hölder functions and unbounded ones which present analytic singularities. The proof is based on a geometric construction of a Bernoulli coding map ω : ( , s, ν) → (P k , f, µ). We obtain the invariance principle for an observable ψ on (P k , f, µ) by applying Philipp-Stout's theorem for χ = ψ • ω on ( , s, ν). The invariance principle implies the central limit theorem as well as several statistical properties for the class U. As an application, we give a direct proof of the absolute continuity of the measure µ when it satisfies Pesin's formula. This approach relies on the central limit theorem for the unbounded observable log Jac f ∈ U.

On the regularity of the Green current for semi-extremal endomorphisms of <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{P}^2 $\end{document}</tex-math></inline-formula>

“…Let us notice that the Central Limit Theorem was used in [27] to prove regularity properties for the equilibrium measure of rational maps on P 1 and in [16] to provide a new proof of the fact that µ is absolutely continuous when f is extremal on P k . Proof.…”

“…Central Limit Theorem. The dynamical system (P 2 , f, µ) satisfies a Central Limit Theorem, see [9,11,12,16]. exists and the following alternative holds:…”

Central limit theorems in linear dynamics

“…We set S n (f ) = n−1 i=0 f • T i and we say that f satisfies the Central Limit Theorem (in short, CLT) if 1 √ n S n (f ) converges in distribution to a normal law. To obtain sufficient conditions on a function f so that the CLT holds, we shall use the martingale method which was successfully used recently in various problems (see for instance [LB99], [CB05], [DS06], [Dup10]). This method goes back to Gordin in [Gor69].…”

Almost sure invariance principle for non-autonomous holomorphic dynamics in ℙ^{𝕜}

“…In particular, the top degree intersection µ f = T k f yields the unique f -invariant measure of maximal entropy ([Lju83, BD01]) with many interesting stochastic properties. For instance, in [Dup10] Dupont obtained an almost sure invariance principle (ASIP) for the holomorphic dynamical system (P k , B, f, µ f ) by using coding techniques and applying Philipp-Stout's theorem [PS75] for observables with analytic singularities. The coding techniques were originally introduced by Przytycki-Urbański-Zdunik [PUZ89] in complex dimension one from which they deduced ASIP (see also [Hay99,PRL07] and the references therein for some statistical results in the case of dimension one).…”