2008/09/16 by Christophe Dupont, Dupont, Christophe
Mathematics · Physics and Astronomy · #37C45 #37F10 #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.0809.2710
openalex publication_date 2008/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let f be an endomorphism of \mathbbCPk and ν be an f-invariant measure with positive Lyapunov exponents (λ1,\...,λk). We prove a lower bound for the pointwise dimension of ν in terms of the degree of f, the exponents of ν and the entropy of ν. In particular our result can be applied for the maximal entropy measure μ. When k=2, it implies that the Hausdorff dimension of μ is estimated by dim\cal H μ≥ log d \over λ1 + log d \over λ2, which is half of the conjectured formula. Our method for proving these results consists in studying the distribution of the ν-generic inverse branches of fn in \mathbbCPk. Our tools are a volume growth estimate for the bounded holomorphic polydiscs in \mathbbCPk and a normalization theorem for the ν-generic inverse branches of fn.