Transcendental Harmonic Mappings and Gravitational Lensing by Isothermal Galaxies

Using the Schwarz function of an ellipse, it was recently shown that galaxies with density constant on confocal ellipses can produce at most four ``bright’’ images of a single source. The more physically interesting example of an isothermal galaxy has density that is constant on \emph{homothetic} ellipses. In that case bright images can be seen to correspond to zeros of a certain transcendental harmonic mapping. We use complex dynamics to give an upper bound on the total number of such zeros.