Parametrized dynamics of the Weierstrass elliptic function

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