Value distribution of derivatives in polynomial dynamics

2019/04/15 by Okuyama, Yûsuke, Vigny, Gabriel
#Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1904.06858

Abstract

For every m∈ℕ, we establish the equidistribution of the sequence of the averaged pull-backs of a Dirac measure at any given value in ℂ∖\0\ under the m-th order derivatives of the iterates of a polynomials f∈ ℂ[z] of degree d>1 towards the harmonic measure of the filled-in Julia set of f with pole at ∞. We also establish non-archimedean and arithmetic counterparts using the potential theory on the Berkovich projective line and the adelic equidistribution theory over a number field k for a sequence of effective divisors on ℙ¹(k) having small diagonals and small heights. We show a similar result on the equidistribution of the analytic sets where the derivative of each iterate of a Hénon-type polynomial automorphism of ℂ² has a given eigenvalue.

Related

• Value distribution of the sequences of the derivatives of iterated polynomials
• Distribution of points with prescribed derivative in polynomial dynamics
• Equidistribution of small points, rational dynamics, and potential theory
• Value Distributions of Derivatives of K-regular Polynomial Families
• Dynamics of rotationally invariant polynomial root sets under iterated differentiations