Dynamics of vertical real rhombic Weierstrass elliptic functions

Abstract

In this paper, we investigate the dynamics of iterating the Weierstrass elliptic functions on vertical real rhombic lattices. The main result of this paper is to show that these functions can have at most one real attracting or parabolic cycle. If there is no real attracting or parabolic cycle, we prove that the real and imaginary axes, as well as translations of these lines by the lattice, lie in the Julia set. Further, we prove that if there exists a real attracting fixed point, then the intersection of the Julia set with the real axis is a Cantor set. Finally, we apply the theorem to find parameters in every real rhombic shape equivalence class for which the Julia set is the entire sphere.