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 $n\rightarrow +\infty$, 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.