Dynamics of singular complex analytic vector fields with essential singularities II

Alvaro Alvarez–Parrilla and Jesús Muciño–Raymundo

Journal of Singularities
volume 24 (2022), 1-78

Received: 16 June 2019. Received in revised form: 3 October 2021.

DOI: 10.5427/jsing.2022.24a

Add a reference to this article to your citeulike library.

Abstract:

Let X be a singular complex analytic vector field on the Riemann sphere described by two polynomials P(z), E(z) of degrees r and d respectively; in such way that X has poles at the roots of P(z), an isolated essential singularity at infinity arising from the exponential of E(z) and no zeros on the complex plane. These vector fields are transcendental of 1–order d. We study the families of the above singular complex analytic vector fields X. For each pair (r,d), with r+d≥1, the family of these vector fields is an open complex manifold of dimension r+d+1. Our goal is the geometric description of the vector fields X, in particular the behaviour of its singularity at infinity. We first exploit that each vector field X has a canonical associated global singular analytic distinguished parameter (the function determined by the integral of the corresponding 1–form of time). Secondly, we develop in full detail the natural one to one correspondence between: vector fields, global singular analytic distinguished parameters and the Riemann surfaces of these distinguished parameters. These Riemann surfaces are biholomorphic to C and have d logarithmic branch points over infinity, d logarithmic branch points over finite asymptotic values and r finitely ramified branch points. As a valuable tool, we introduce (r,d)–configuration trees, which are weighted directed rooted trees. An (r,d)–configuration tree completely encodes the Riemann surface of a vector field X, including its associated singular flat metric.

Our main result states that the (r,d)–configuration trees provide local holomorphic parameters, for the corresponding family of vector fields. Explicit geometrical and dynamical information is supplied by (r,d)–configuration trees. In fact, given a vector field in the family, the phase portrait of the associated real vector field on the Riemann sphere is decomposed into real flow invariant regions, half planes and strips. The structural stability (under perturbation in the corresponding family) of the phase portrait of the real vector field, is characterized by using (r,d)–configuration trees. The number of (orientation preserving) topologically classes of real phase portraits is counted in terms of certain conditions of the discrete parameters (r,d). The germ of the isolated essential singularity of the vector field X is described as a combinatorial word consisting of hyperbolic, elliptic, parabolic and (suitable) entire angular sectors at infinity. Our work has its roots in the seminal study of R. Nevanlinna.

Author(s) information:

Alvaro Alvarez–Parrilla
Grupo Alximia SA de CV, Ensenada
Baja California
CP 22800 México
email: alvaro.uabc@gmail.com

Jesús Muciño–Raymundo
Centro de Ciencias Matemáticas
Universidad Nacional Autónoma de México
Morelia, México
email: muciray@matmor.unam.mx