Newton flows for elliptic functions I Structural stability: Characterization & Genericity

TL;DR

This paper develops a theory of elliptic Newton flows, showing that for most elliptic functions of fixed order, the associated dynamical systems are structurally stable and can be characterized by nondegeneracy conditions.

Contribution

It extends the theory of Newton flows to elliptic functions, providing a characterization and demonstrating generic structural stability under perturbations.

Findings

Almost all elliptic functions of fixed order have structurally stable Newton flows.

Structural stability is characterized by nondegeneracy properties of the functions.

The theory parallels the rational case, extending understanding of elliptic Newton flows.

Abstract

Newton flows are dynamical systems generated by a continuous, desingularized Newton method for mappings from a Euclidean space to itself. We focus on the special case of meromorphic functions on the complex plane. Inspired by the analogy between the rational (complex) and the elliptic (i.e., doubly periodic meromorphic) functions, a theory on the class of so-called Elliptic Newton flows is developed. With respect to an appropriate topology on the set of all elliptic functions $f$ of fixed order $ r (\geqslant 2)$ we prove: For almost all functions $f$, the corresponding Newton flows are structurally stable i.e., topologically invariant under small perturbations of the zeros and poles for $f$ [ genericity]. They can be described in terms of nondegeneracy-properties of $f$ similar to the rational case [characterization].