Dynamics of singular complex analytic vector fields with essential singularities I

Abstract

We tackle the problem of understanding the geometry and dynamics of singular complex analytic vector fields $X$ with essential singularities on a Riemann surface $M$ (compact or not). Two basic techniques are used. (a) In the complex analytic category on $M$, we exploit the correspondence between singular vector fields $X$, differential forms $\omega _{X}$ (with $\omega _{X}(X)\equiv 1$), orientable quadratic differentials $\omega _{X} \otimes \omega _{X}$, global distinguished parameters $\Psi _{X} (z) = \int ^z \omega _{X}$, and the Riemann surfaces $\mathcal {R}_{X}$ of the above parameters. (b) We use the fact that all singular complex analytic vector fields can be expressed as the global pullback via certain maps of the holomorphic vector fields on the Riemann sphere, in particular, via their respective $\Psi _{X}$. We show that under certain analytical conditions on $\Psi _{X}$, the germ of a singular complex analytic vector field determines a decomposition in angular sectors; center $C$, hyperbolic $H$, elliptic $E$, parabolic $P$ sectors but with the addition of suitable copies of a new type of entire angular sector $\mathscr {E}$, stemming from $X(z)=\mathrm{e}^z \frac {\partial }{\partial z}$. This extends the classical theorems of A. A. Andronov et al. on the decomposition in angular sectors of real analytic vector field germs. The Poincaré–Hopf index theory for $\mathfrak {Re}\left (X\right )$ local and global on compact Riemann surfaces, is extended so as to include the case of suitable isolated essential singularities. The inverse problem: determining which cyclic words $\mathcal {W}_{X}$, comprised of hyperbolic, elliptic, parabolic and entire angular sectors, it is possible to obtain from germs of singular analytic vector fields, is also answered in terms of local analytical invariants. We also study the problem of when and how a germ of a singular complex analytic vector field having an essential singularity (not necessarily isolated) can be extended to a suitable compact Riemann surface. Considering the family of entire vector fields $\mathcal {E}(d) =\{X(z)= \lambda \mathrm{e}^{P(z)}\frac {\partial }{\partial z}\}$ on the Riemann sphere, where $P(z)$ is a polynomial of degree $d$ and $\lambda \in \mathbb {C}^*$, we completely characterize the local and global dynamics of this class of vector fields, compute analytic normal forms for $d=1, 2, 3$, and show that for $d\geq 3$ there are an infinite number of topological (phase portrait) classes of $\mathfrak {Re}(X)$, for $X\in \mathcal {E}(d)$. These results are based on the work of R. Nevanlinna, A. Speisser and M. Taniguchi on entire functions $\Psi _{X}$. Finally, on the topological decomposition of real vector fields into canonical regions, we extend the results of L. Markus and H. E. Benzinger to meromorphic on $\mathbb {C}$ vector fields $X$, with an essential singularity at $\infty \in \widehat {\mathbb {C}}$, whose $\Psi _{X}^{-1}$ have $d$ logarithmic branch points over $d$ finite asymptotic values and $d$ logarithmic branch points over $\infty$.