Bernoulli coding map and almost sure invariance principle for endomorphisms of Pk

• 成果类型: Article
• 署名作者: Dupont, Christophe
• 署名单位: Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-008-0192-4
• 发表日期: 2010
• 页码: 337-359
• 关键词: CENTRAL-LIMIT-THEOREM; dimension

摘要

Let f be an holomorphic endomorphism of P-k and mu be its measure of maximal entropy. We prove an almost sure invariance principle for the systems (P-k, f, mu). Our class U of observables includes the Holder 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) -> (P-k, f, mu). We obtain the invariance principle for an observable psi on (P-k, f, mu) by applying Philipp-Stout's theorem for chi = psi omicron omega on (Sigma, s, nu). 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 mu when it satisfies Pesin's formula. This approach relies on the central limit theorem for the unbounded observable log Jac f is an element of U.