2016/09/05 by G. F. Helminck, Helminck, G. F., F. Twilt +1
Mathematics · #30C15 #30D30 #30F99 #33E05 #34D30 #37C15 #37C20 #37C70 #49M15 #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Equations Stability Results #Mathematical Dynamics and Fractals #math.CV #math.DS #msc:30C15 #msc:30D30 #msc:30F99 #msc:33E05 #msc:34D30 #msc:37C15 #msc:37C20 #msc:37C70 #msc:49M15
paper · pdf · doi:10.48550/arxiv.1609.01267
21 pages, 11 figures, references added in sections 1,2,4 and 5, typos corrected, arguments and relevance clarified
openalex publication_date 2016/09/05 · openalex created_date 2016/09/16 · arxiv created 2017/03/21 · arxiv updated 2017/03/22 · openalex updated_date 2026/07/28
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].