Value Distributions of Derivatives of $K$-regular Polynomial Families

Abstract

Let $\Omega \in \mathbb{C}$ be a domain such that $K:=\mathbb{C}\setminus \Omega$ is compact and non-polar. Let $g_{\Omega }$ be the Green's function with a logarithmic pole at infinity, and let $\omega ={\omega }_K$ be the equilibrium distribution on $K$. Let $(q_k{)}_{k>0}$ be a sequence of polynomials with $n_k$, the degree of $q_k$ satisfying $n_k\to \infty$, and let $(q_k^m{)}_k$ denote the sequence of $m$-th derivatives. We provide conditions, which ensure that the preimages $(q_k^m{)}^{-1}(\{a\})$ uniformly equidistribute on $\partial \Omega$, as $k\to \infty$, for every $a\in \mathbb{C}$ and every $m=0,1,\dots$

References in corpus (1)

• Equidistribution of the zeros of higher order derivatives in polynomial dynamics