Parametrized dynamics of the Weierstrass elliptic function

Abstract

We study parametrized dynamics of the Weierstrass elliptic ℘ \wp function by looking at the underlying lattices; that is, we study parametrized families ℘ Λ \wp _{\Lambda } and let Λ \Lambda vary. Each lattice shape is represented by a point τ \tau in a fundamental period in modular space; for a fixed lattice shape Λ = [ 1 , τ ] \Lambda = [1, \tau ] we study the parametrized space k Λ k \Lambda . 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 ( g 2 , g 3 ) (g_2, g_3) to parametrize some lattices.