Value distribution of the sequences of the derivatives of iterated polynomials

2016/07/27 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.1607.08037

openalex publication_date 2016/07/27 · openalex created_date 2016/08/23 · openalex updated_date 2026/07/28

Abstract

We establish the equidistribution of the sequence of the averaged pullbacks of a Dirac measure at any value in ℂ∖\0\ under the derivatives of the iterations of a polynomials f∈ℂ[z] of degree more than one towards the f-equilibrium (or canonical) measure μ_f on ℙ¹. We also show that for every C² test function on ℙ¹, the convergence is exponentially fast up to a polar subset of exceptional values in ℂ. A parameter space analog of the latter quantitative result for the monic and centered unicritical polynomials family is also established.

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