Geometric formulas on Rumely's weight function and crucial measure in non-archimedean dynamics

2016/12/14 by Yûsuke Okuyama, Okuyama, Yûsuke
Mathematics · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1612.04441

openalex publication_date 2016/12/14 · openalex created_date 2017/01/06 · openalex updated_date 2026/07/28

Abstract

We introduce the f-crucial function Crucial_f associated to a rational function f∈ K(z) of degree >1 over an algebraically closed field K of possibly positive characteristic that is complete with respect to a non-trivial and non-archimedean absolute value, and give a global and explicit expression of Rumely's (resultant) function ordRes_f in terms of the hyperbolic metric ρ on the Berkovich upper half space H¹ in the Berkovich projective line P=P(K). We also obtain geometric formulas for Rumely's weight function w_f and crucial measure ν_f on P¹ associated to f, as well as improvements of Rumely's principal results. As an application to dynamics, we obtain a quantitative equidistribution of the sequence (ν_fⁿ)_n of fⁿ-crucial measures towards the f-equilibrium (or canonical) measure μ_f on P¹.

Citations

• Iteration and the Minimal Resultant

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