Discrete conformal approximation of complex earthquake maps

Using the techniques of circle packing, we construct discrete conformal approximations for complex earthquake maps on the Teichmüller spaces of compact, hyperbolic Riemann surfaces developed by William Thurston and Curtis McMullen, and we show that these approximations are convergent. We then describe earthquake maps on the Teichmüller spaces of compact, Euclidean Riemann surfaces, extending the work of Thurston and McMullen. Using the discrete conformal approximations developed for hyperbolic surfaces, we approximate the action of these new maps with circle packing.