Toral band Fatou components for the Weierstrass ℘ function

Abstract

We study the existence of unbounded Fatou components for the iterated Weierstrass ℘ \wp function and discuss its dependence on lattice shapes. We show that there are open regions in shape space for lattices in C \mathbb {C} for which scalings of the lattice (real and complex) yield unbounded toral band Fatou components.