Zeros of Faber Polynomials for Joukowski Airfoils

Abstract

Let K be the closure of a bounded region in the complex plane with simply connected complement whose boundary is a piecewise analytic curve with at least one outward cusp. The asymptotics of zeros of Faber polynomials for K are not understood in this general setting. Joukowski airfoils provide a particular class of such sets. We determine the (unique) weak-* limit of the full sequence of normalized counting measures of the Faber polynomials for Joukowski airfoils; it is never equal to the potential-theoretic equilibrium measure of K. This implies that many of these airfoils admit an electrostatic skeleton and also explains an interesting class of examples of Ullman [11] related to Chebyshev quadrature.