Zeros of the Derivatives of Faber Polynomials Associated with a Universal Covering Map

Abstract

For a compact set $E \subset \mathbb{C}$ containing more than two points, we study asymptotic behavior of normalized zero counting measures $\{\mu_k \}$ of the derivatives of Faber polynomials associated with $E$. For example if $E$ has empty interior, we prove that $\{\mu_k \}$ converges in the weak-star topology to a measure whose support is the boundary of $g(\mathcal{D})$, where $g : \{|z| > r \}\cup \{\infty\} \to \bar{\mathbb{C}} \backslash E$ is a universal covering map such that $g(\infty) = \infty$ and $\mathcal{D}$ is the Dirichlet domain associated with $g$ and centered at $\infty$. Our results are counterparts of those of Kuijlaars and Saff (1995) on zeros of Faber polynomials.