An a priori bound of endomorphisms of ℂℙ^k and a remark on the Makienko conjecture in dimension one

2018/06/19 by Yûsuke Okuyama, Okuyama, Yûsuke
Mathematics · #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1806.07152

openalex publication_date 2018/06/19 · openalex created_date 2018/06/29 · openalex updated_date 2026/07/28

Abstract

Let f be an endomorphism of ℂℙ^k of degree >1, and assume that for any cyclic Fatou component W of f having a period p∈ℕ, the equilibrium measure μ_f has a positive charge on the boundary of W if and only if f^-p(W)=W. Then we obtain a locally uniform a priori bound of the dynamics of f, which in particular yields a Diophantine-type estimate of the dynamics of f on its domaines singuliers. We also point out that in the case of k=1, the statement of our assumption is related to both the impossibility for the Julia set of f to be the boundary of lakes of Wada and the so called Makienko conjecture on the non-emptiness of the residual Julia set of f.

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