Dynamics of singular complex analytic vector fields with essential singularities II

TL;DR

This paper geometrically characterizes singular complex analytic vector fields with essential singularities at infinity, using configuration trees to encode their structure and stability, and describes their phase portraits and perturbation behavior.

Contribution

It introduces $(r,d)$-configuration trees that encode the geometry and dynamics of these vector fields, providing explicit tools for their analysis and stability characterization.

Findings

Configuration trees encode Riemann surface and metric structure.

Phase portraits decompose into invariant components like half-planes.

Stability criteria depend explicitly on parameters $r$ and $d$.

Abstract

The singular complex analytic vector fields $X$ on the Riemann sphere $\widehat{\mathbb C}_z$ belonging to the family ${\mathscr E}(r,d)=\left\{ X(z)=\frac{1}{P(z)} e^{E(z)}\frac{\partial }{\partial z}\ \Big\vert \ P, E\in\mathbb{C}[z]\right\}$, where $P$ is monic, $deg(P)=r$, $deg(E)=d$, $r+d\geq 1$, have a finite number of poles on the complex plane and an isolated essential singularity at infinity (for $d\geq 1$). Our aim is to describe geometrically $X$, particularly the singularity at infinity. We use the natural one to one correspondence between $X$, a global singular analytic distinguished parameter $\Psi_X(z)=\int^z P(\zeta) e^{-E(\zeta)}d\zeta$, and the Riemann surface ${\mathcal R}_X$ of this distinguished parameter. We introduce $(r,d)$-configuration trees which are weighted directed rooted trees. An $(r,d)$-configuration tree completely encodes the Riemann surface ${\mathcal R}_X$ and the singular flat metric associated on ${\mathcal R}_X$. The $(r,d)$-configuration trees provide "parameters" for the complex manifold ${\mathscr E}(r,d)$, which give explicit geometrical and dynamical information; a valuable tool for the analytic description of $X\in{\mathscr E}(r,d)$. Furthermore, given $X$, the phase portrait of the associated real vector field $Re(X)$ on the Riemann sphere is decomposed into $Re(X)$-invariant components: half planes and finite height strips. The germ of $X$ at infinity is described as a combinatorial word (consisting of hyperbolic, elliptic, parabolic and entire angular sectors having the point at infinity of $\widehat{\mathbb C}_z$ as center). The structural stability, under perturbation in ${\mathscr E}(r,d)$, of the phase portrait of $Re(X)$ is characterized by using the $(r,d)$-configuration trees. We provide explicit conditions, in terms of $r$ and $d$, as to when the number of topologically equivalent phase portraits of $Re(X)$ is unbounded.