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

Abstract

Let $f$ be an holomorphic endomorphism of $\mathbb{P}^k$ and $μ$ be its measure of maximal entropy. We prove an Almost Sure Invariance Principle for the systems $(\mathbb{P}^k,f,μ)$. Our class $\cal{U}$ 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 $ω: (Σ, s, ν) \to (\mathbb{P}^k,f,μ)$. We obtain the invariance principle for an observable $ψ$ on $(\mathbb{P}^k,f,μ)$ by applying Philipp-Stout's theorem for $χ= ψ\circ ω$ on $(Σ, s, ν)$. 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 $μ$ when it satisfies Pesin's formula. This approach relies on the Central Limit Theorem for the unbounded observable $\log \textsf{Jac} f \in \cal{U}$.