Equidistribution of the zeros of higher order derivatives in polynomial dynamics

TL;DR

This paper proves that the zeros of higher order derivatives of iterated polynomials become equidistributed according to the harmonic measure of the Julia set, extending previous results and using advanced functional equation techniques.

Contribution

It establishes the convergence of zeros of higher derivatives of polynomial iterates to the harmonic measure of the Julia set, providing new insights into polynomial dynamics and zero distribution.

Findings

Zeros of higher derivatives converge to harmonic measure

Extension of previous zero distribution results

Uses approximation of functional equations for proofs

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 Schr\"oder or Abel functional equations for a meromorphic function on $\mathbb{C}$ with a locally uniform non-trivial error estimate.