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Equidistribution of the zeros of higher order derivatives in polynomial dynamics

Yûsuke Okuyama

Abstract

For every $m\in\mathbb{N}$, we establish the convergence of the averaged distributions of the zeros of the $m$-th order derivatives $(f^n)^{(m)}$ of the iterated polynomials $f^n$ of a polynomial $f\in\mathbb{C}[z]$ of degree $>1$ towards the harmonic measure of the filled-in Julia set of $f$ with pole at $\infty$ as $n\to+\infty$, when $f$ has no exceptional points in $\mathbb{C}$. This complements our former study on the zeros of $(f^n)^{(m)}-a$ for any value $a\in\mathbb{C}\setminus\{0\}$. The key in the proof is an approximation of the higher order derivatives of a solution of the Schröder or Abel functional equations for a meromorphic function on $\mathbb{C}$ with a locally uniform non-trivial error estimate.

Equidistribution of the zeros of higher order derivatives in polynomial dynamics

Abstract

For every , we establish the convergence of the averaged distributions of the zeros of the -th order derivatives of the iterated polynomials of a polynomial of degree towards the harmonic measure of the filled-in Julia set of with pole at as , when has no exceptional points in . This complements our former study on the zeros of for any value . The key in the proof is an approximation of the higher order derivatives of a solution of the Schröder or Abel functional equations for a meromorphic function on with a locally uniform non-trivial error estimate.
Paper Structure (7 sections, 3 theorems, 41 equations)

This paper contains 7 sections, 3 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['th:convergence']}
  3. Schröder and Abel functional equations
  4. Preparatory computations
  5. Proof of Theorem \ref{['th:anyorder']}
  6. Observation from OVderivative
  7. Further observation using complex dynamics

Key Result

Theorem 1

Let $f\in\mathbb{C}[z]$ be a polynomial of degree $d>1$, and $m\in\mathbb{N}$. If $E(f)=\{\infty\}$, then

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • proof : Proof of Theorem \ref{['th:convergence']}
  • Theorem 3
  • proof : Proof of Theorem \ref{['th:potential']}