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 $\mu$ be its measure of maximal entropy. We prove an Almost Sure Invariance Principle for the systems $({\mathbb{P}}^k,f,\mu )$. Our class $\mathcal{U}$ 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 $\omega :(\Sigma ,s,\nu )\to ({\mathbb{P}}^k,f,\mu )$. 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 $\mathcal{U}$. 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 $\log \text{Jac}f\in \mathcal{U}$.

25 pages, to appear in Probability Theory and Related Fields

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