Equidistribution of the zeros of higher order derivatives in polynomial dynamics

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.

7 pages

Cited by in corpus (2)

• Value Distributions of Derivatives of $K$-regular Polynomial Families
• Convergence of Equilibrium Measures under $K$-regular Polynomial Sequences and their Derivatives