Zeros of the derivatives of Faber polynomials associated with a universal covering map

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