Bernoulli coding map and almost sure invariance principle for endomorphisms of ℙ^k

2007/12/04 by Christophe Dupont, Dupont, Christophe
Mathematics · #37C40 #37F10 #60F17 #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #math.CV #math.DS #msc:37C40 #msc:37F10 #msc:60F17

paper · pdf · doi:10.48550/arxiv.0712.0521

25 pages, to appear in Probability Theory and Related Fields

arxiv created 2008/12/06 · arxiv updated 2009/12/01

Abstract

Let f be an holomorphic endomorphism of ℙ^k and μ be its measure of maximal entropy. We prove an Almost Sure Invariance Principle for the systems (ℙ^k,f,μ). Our class \calU 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) → (ℙ^k,f,μ). We obtain the invariance principle for an observable ψ on (ℙ^k,f,μ) by applying Philipp-Stout's theorem for χ= ψ∘ ω on (Σ, s, ν). The invariance principle implies the Central Limit Theorem as well as several statistical properties for the class \calU. As an application, we give a 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 \textsfJac f ∈ \calU.

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