Dynamics of singular complex analytic vector fields with essential singularities II

Abstract

Generically, the singular complex analytic vector fields $X$ on the Riemann sphere $\widehat{\mathbb{C}}_{z}$ belonging to the family $$ \mathscr{E}(r,d)=\Big\{ X(z)=\frac{1}{P(z)}\ \text{e}^{E(z)}\frac{\partial}{\partial z} \ \big\vert \ P, E\in\mathbb{C}[z], \ deg(P)=r, \ deg(E)=d \Big\}, $$ have an essential singularity of finite 1-order at infinity and a finite number of poles on the complex plane. We describe $X$, particularly the singularity at $\infty\in\widehat{\mathbb{C}}_{z}$. In order to do so, we use the natural $correspondence$ between $X\in\mathscr{E}(r,d)$, a global singular analytic distinguished parameter $\Psi_X=\int \omega_X$, and the Riemann surface $\mathcal{R}_X$ of the distinguished parameter. We introduce $(r,d)$-$configuration\ trees$ $\Lambda_X$: combinatorial objects that completely encode the Riemann surfaces $\mathcal{R}_X$ and singular flat metrics associated to $X\in\mathscr{E}(r,d)$. This provides an alternate `dynamical' coordinate system and an analytic classification of $\mathscr{E}(r,d)$. Furthermore, the phase portrait of $\mathscr{Re}(X)$ on $\mathbb{C}$ is decomposed into $\mathscr{Re}(X)$-invariant regions: half planes and finite height strip flows. The germ of $X$ at $\infty \in \widehat{\mathbb{C}}$ is described as an admissible word (equivalent to certain canonical angular sectors). The structural stability of the phase portrait of $\mathscr{Re}(X)$ is characterized by using $\Lambda_X$ and the number of topologically equivalent phase portraits of $\mathscr{Re}(X)$ is bounded.