Parameterized Dynamics for the Weierstrass Elliptic Function over Square Period Lattices

We iterate the Weierstrass elliptic ℘ function in order to understand the dependence of the dynamics on the underlying period lattice L. We focus on square lattices and use the holomorphic dependence on the classical invariants (g2, g3) = (g2, 0) to show that in parameter space (g2-space) one sees both quadratic-like attracting orbit behavior and pre-pole dynamics. In the case of pre-pole parameters all critical orbits terminate at poles and the Julia set of ℘L is the entire sphere. We show that both the Mandelbrot-like dynamics and the pre-pole parameters accumulate on pre-pole parameters of lower order providing results on the dynamics occurring in parameter space “between Mandelbrot sets”.