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

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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 \({(\mathbb{P}^k,f,\mu)}\). 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 ψ on \({(\mathbb{P}^k,f,\mu)}\) by applying Philipp–Stout’s theorem for \({\chi = \psi \circ \omega}\) on (Σ, s, ν). 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 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 \({{\tt Jac}\, f \in \mathcal{U}}\).

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Correspondence to Christophe Dupont.

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Dupont, C. Bernoulli coding map and almost sure invariance principle for endomorphisms of \({\mathbb P^k}\) . Probab. Theory Relat. Fields 146, 337–359 (2010). https://doi.org/10.1007/s00440-008-0192-4

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Keywords

  • Holomorphic dynamics
  • Bernoulli coding map
  • Almost sure invariance principle

Mathematics Subject Classification (2000)

  • 37F10
  • 37C40
  • 60F17