Parameterized Dynamics for the Weierstrass Elliptic Function Over Square Period Lattices

Abstract

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 prepole dynamics. In the case of prepole 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 prepole parameters accumulate on prepole parameters of lower order providing results on the dynamics occurring in parameter space "between Mandelbrot sets".

Proof of a folklore Julia set connectedness theorem and connections with elliptic functions

“…and (2) follows easily by multiplying equations (13) and (14). To prove (3), we note that by Proposition 2.1,…”

“…We also make connections between the iteration of elliptic functions and their truncated Laurent series, noting that connectivity results about Julia sets of Weierstrass elliptic ℘ functions over certain period lattices are proved in [4,5,10,11,13] and studies of their parameter spaces appear in [13,14]. The techniques used in these papers, along with studies done on the maps of the form (1), studied copiously by Devaney and other authors (for example, [3,[6][7][8]) lead to a natural proof of this theorem.…”

“…However making the right adjustment reveals many similarities between the families, including at the level of parameter space (compare, e.g., [7], [12], and [14]). …”