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

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 $g:\{\mathrm{\mid }z\mathrm{\mid }>r\}\cup \{\mathrm{\infty }\}\to \bar{\mathbb{C}}\mathrm{\backslash }E$ is a universal covering map such that $g(\mathrm{\infty })=\mathrm{\infty }$ and $\mathcal{D}$ is the Dirichlet domain associated with $g$ and centered at $\mathrm{\infty }$. Our results are counterparts of those of Kuijlaars and Saff (1995) on zeros of Faber polynomials.