---
abstract: |
    Let $f$ be an holomorphic endomorphism of $\mathbb{P}^k$ and $\mu$ be its measure of maximal entropy. We prove an Almost Sure Invariance Principle for the systems $(\mathbb{P}^k,f,\mu)$. Our class $\UU$ 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 $\omega: (\Sigma , s , \nu) \to
    (\mathbb{P}^k,f,\mu)$. We obtain the invariance principle for an observable $\psi$ on $(\mathbb{P}^k,f,\mu)$ by applying Philipp-Stout’s theorem for $\chi = \psi
    \circ \omega$ on $(\Sigma , s , \nu)$. The invariance principle implies the Central Limit Theorem as well as several statistical properties for the class $\UU$. As an application, we give a *direct* proof of the absolute continuity of the measure $\mu$ when it satisfies Pesin’s formula. This approach relies on the Central Limit Theorem for the unbounded observable $\log
    \textsf{Jac} \,  f \in \UU$.
author:
- 'C. Dupont'
date:  
title: 'Bernoulli coding map and almost sure invariance principle for endomorphisms of ${\mathbb {P}}^k$'
---

Introduction {#intro}
============

Let $f : {\mathbb {P}}^k \to {\mathbb {P}}^k$ be an holomorphic endomorphism of algebraic degree $d
\geq 2$. Its equilibrium measure $\mu$ is the limit of 