Walsh's conformal map onto lemniscatic domains for polynomial pre-images I

Abstract

We consider Walsh's conformal map from the exterior of a compact set $E \subseteq \mathbb{C}$ onto a lemniscatic domain. If $E$ is simply connected, the lemniscatic domain is the exterior of a circle, while if $E$ has several components, the lemniscatic domain is the exterior of a generalized lemniscate and is determined by the logarithmic capacity of $E$ and by the exponents and centers of the generalized lemniscate. For general $E$, we characterize the exponents in terms of the Green's function of $E^c$. Under additional symmetry conditions on $E$, we also locate the centers of the lemniscatic domain. For polynomial pre-images $E = P^{-1}(\Omega)$ of a simply-connected infinite compact set $\Omega$, we explicitly determine the exponents in the lemniscatic domain and derive a set of equations to determine the centers of the lemniscatic domain. Finally, we present several examples where we explicitly obtain the exponents and centers of the lemniscatic domain, as well as the conformal map.