Dynamics of Newton's method for odd and even elliptic functions

TL;DR

This paper studies the dynamics of Newton's method when applied to odd or even elliptic functions with specific pole and lattice conditions, revealing connected Julia sets and conditions for wandering domains.

Contribution

It provides new insights into the dynamics of Newton's method on elliptic functions, including conditions for connected Julia sets and coexistence of wandering domains and attracting basins.

Findings

Julia set of Newton map is connected if no Herman rings exist.

Sufficient conditions for wandering domains coexistence with attracting basins.

Newton maps of certain elliptic functions lack Herman rings or Baker domains, with bounded Fatou components.

Abstract

We investigate Newton's method applied to any odd or any even elliptic function with an arbitrary period lattice. For any function of this type whose set of poles coincides with its period lattice, we show that the Julia set of its Newton map is connected, as long as no Herman rings exist. Moreover, we provide sufficient conditions on the Newton's method of any odd or even elliptic function to exhibit wandering domains coexisting with attracting basins. This phenomenon was first reported by Florido and Fagella for the Newton's method applied to a family of entire functions; however, our approach does not employ the lifting technique. We also provide a detailed study of a one-parameter family of elliptic functions given by $\wp_\Lambda+b$ with $\Lambda$ any triangular period lattice and $b\in\mathbb{C}$. We show that their associated Newton maps do not exhibit Herman rings or Baker domains, and any other Fatou component, including wandering, is bounded.