Parametrized dynamics of the weierstrass elliptic function

Abstract

We study parametrized dynamics of the Weierstrass elliptic ϑ function by looking at the underlying lattices; that is, we study parametrized families ϑ Ʌ and let Ʌ vary. Each lattice shape is represented by a point τ in a fundamental period in modular space; for a fixed lattice shape Ʌ = [1, τ] we study the parametrized space kɅ. We show that within each shape space there is a wide variety of dynamical behavior, and we conduct a deeper study into certain lattice shapes such as triangular and square. We also use the invariant pair (g2; g3) to parametrize some lattices. © 2004 American Mathematical Society.