Dynamics of singular complex analytic vector fields with essential singularities II

Alvaro Alvarez–Parrilla ; Jesús Muciño–Raymundo

Journal of Singularities, Tome 24 (2022), pp. 1-78

Résumé

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.

Détail

DOI : 10.5427/jsing.2022.24a