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Conformal Geometry and Dynamics

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Dynamics of singular complex analytic vector fields with essential singularities I


Authors: Alvaro Alvarez–Parrilla and Jesús Muciño–Raymundo
Journal: Conform. Geom. Dyn. 21 (2017), 126-224
MSC (2010): Primary 32S65; Secondary 30F20, 58K45, 32M25
DOI: https://doi.org/10.1090/ecgd/306
Published electronically: March 16, 2017
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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)=$$ \text {\Large 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$$ \text {\Large 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 $.


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Additional Information

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

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

DOI: https://doi.org/10.1090/ecgd/306
Keywords: Entire functions, entire vector fields, Riemann surfaces, essential singularities
Received by editor(s): February 20, 2014
Received by editor(s) in revised form: July 8, 2016, July 29, 2016, and December 14, 2016
Published electronically: March 16, 2017
Additional Notes: The first author was partially supported by UABC projects 1273 and 0196
The second author was partially supported by LAISLA
Dedicated: Dedicated to Luz Ximena; she loves the pictures.
Article copyright: © Copyright 2017 American Mathematical Society