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We apply the charge simulation method (CSM) in order to compute the logarithmic capacity of compact sets consisting of (infinitely) many “small” components. This application allows to use just a single charge point for each component. The resulting method therefore is significantly more efficient than methods based on discretizations of the boundaries (for example, our own method presented in Liesen et al. (Comput. Methods Funct. Theory 17, 689–713, 2017)), while maintaining a very high level of accuracy. We study properties of the linear algebraic systems that arise in the CSM, and show how these systems can be solved efficiently using preconditioned iterative methods, where the matrix-vector products are computed using the fast multipole method. We illustrate the use of the method on generalized Cantor sets and the Cantor dust.
We consider Walsh’s conformal map from the exterior of a compact set E ⊆ 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 Ec. Under additional symmetry conditions on E, we also locate the centers of the lemniscatic domain. For polynomial pre-images E = P−1(Ω) of a simply-connected infinite compact set Ω, 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.