Antiholomorphic perturbations of Weierstrass Zeta functions and Green's function on tori

TL;DR

This paper analyzes the parameter space of antiholomorphic perturbations of Weierstrass Zeta functions on tori, providing a topological classification of hyperbolic regions and confirming a conjecture about critical point constancy of Green's functions.

Contribution

It offers a complete topological description of hyperbolic components and their boundaries, and introduces natural parametrizations by dynamical invariants, settling a prior conjecture.

Findings

Topological classification of hyperbolic components

Boundary descriptions of parameter spaces

Confirmation of a conjecture on critical point regions

Abstract

In \cite{BeEr}, Bergweiler and Eremenko computed the number of critical points of the Green's function on a torus by investigating the dynamics of a certain family of antiholomorphic meromorphic functions on tori. They also observed that hyperbolic maps are dense in this family of meromorphic functions in a rather trivial way. In this paper, we study the parameter space of this family of meromorphic functions, which can be written as antiholomorphic perturbations of Weierstrass Zeta functions. On the one hand, we give a complete topological description of the hyperbolic components and their boundaries, and on the other hand, we show that these sets admit natural parametrizations by associated dynamical invariants. This settles a conjecture, made in \cite{LW}, on the topology of the regions in the upper half plane $\mathbb{H}$ where the number of critical points of the Green's function remains constant.