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

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 $\cal{U}$ of observables includes the H\"older 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 $\cal{U}$. As an application, we give a \emph{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 $\log \textsf{Jac} f \in \cal{U}$.