Zeros of Faber polynomials for Joukowski airfoils (Q784749)

The paper is devoted to the problem of the asymptotics of zeros of Faber polynomials for \(K \subset \mathbb{C}\), where \(K\) is the closure of a bounded region with simply connected complement \(\overline{\mathbb{C}}\backslash K\) whose boundary is a piecewise analytic curve with one outward gasp. The authors investigate a particular case of such sets provided by Joukowski airfoils. They consider \(\Psi=J\circ T:\ \{z: |z|>1\} \to \mathbb{C}\backslash K\), where \(J(\zeta)=\frac{1}{2}\left(\zeta+\frac{1}{\zeta}\right)\) and \(\zeta=T(z)=R e^{i\theta}(z-1)+1\) with \(R>1\) and \(\theta \in (-\pi/2, \pi/2)\). If \(\Phi = \Psi^{-1}\), then the Faber polynomials for \(K\) are such polynomials \(\{F_n\}\) that \(F_n(z)=(\Phi(z))^n+\mathcal{O}(1/z)\), \(z\to \infty\). Based on the explicit formula \(F_n(z)=a^{-n} \left[ \left(z+(-b)+\sqrt{ z^2-1}\right)^n+\left(z+(-b)-\sqrt{ z^2-1}\right)^n-(-b)^n\right]\), where \(a=Re^{i\theta}\) and \(b=1-Re^{i\theta}\), the authors find the following asymptotics of the zeros \(z_1^{(n)}, \ldots, z_n^{(n)}\) of \(F_n:\) 1. If \(\theta=0\) for \(1<R\leq 3/2\), all zeros of \(F_n\) lie in \([-1,1];\) and for \(R>3/2\) they accumulate on \([1/2b,1]\cup\mathcal{L}^{+}\), where \( \mathcal{L}^{+} =\{z: |b-z-\sqrt{z^2-1}|=|b|\}\). 2. For \(1<R \cos\theta \leq 3/2\), all zeros of \(F_n\) approach \(\mathcal{A}=\left \{z: \frac{z-b}{\sqrt{b^2+1-2bz}} \in [-1,1]\right \}\); and for \(R \cos\theta> 3/2\), they accumulate on \(\mathcal{A}_b\cup \mathcal{L}^{+}_b\), where \(\mathcal{A}_b\) and \(\mathcal{L}^{+}_b\) are specific portions of corresponding curves \(\mathcal{A}\) and \(\mathcal{L}^{+}\). The authors also determine the unique weak-* limit of the full sequence of normalized counting measures \(\mu_n:=\frac{1}{n}\sum _{j=1}^{n} \delta_{ z_j^{(n)}}\) of \(F_n\). They show that the limit is never equal to the potential-theoretic equilibrium measure of \(K\). This implies that these airfoils admit an electrostatic skeleton and also explains an interesting class of examples of \textit{J. L. Ullman} [Mich. Math. J. 13, 417--423 (1966; Zbl 0152.25401)] related to Chebyshev quadrature.