Bifurcations of meromorphic vector fields on the Riemann sphere

Muciño–Raymundo, Jesús; Valero, Carlos

doi:10.1017/s0143385700009883

Cited by 20 publications

(34 citation statements)

References 8 publications

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“…Summing up we get the following result (first introduced for (1)(2)(3)(4) in [50], [49], for (5) in the rational case [17], [18] and now expanded to cover (6) …”

Section: Definition 24 1 a Global Analytic Function ξ Is A Collectmentioning

confidence: 93%

“…The point of view of differential equations (meromorphic vector fields): J. Gregor [26], [27], O. Hájek [31], [32], [33], N. A. Lukashevich [46], L. Brickman et al [14], M. Sabatini [59], J. Muciño-Raymundo et al [50], D. J. Needhan et al [52], E. P. Volokitin et al [70], A. Alvarez-Parrilla et al [4], A. Garijo et al [24], B. Branner et al [13], E. Frías-Armenta et al [23].…”

Section: E(d)mentioning

confidence: 99%

“…Partial versions of the correspondence (1)-(6): J. Muciño-Raymundo et al [50], J. Muciño-Raymundo [49], A. Bustinduy et al [17], [18].…”

Section: E(d)mentioning

confidence: 99%

“…Flat metrics: K. Strebel [66], R. Peretz [57], J. Muciño-Raymundo et al [50], H. Masur et al [48], J. Muciño-Raymundo [49], M. Kontsevich et al [44], J. P. Bowman et al [15].…”

Section: E(d)mentioning

confidence: 99%

“…Notation and conventions. The material covered in this section can be found, for the meromorphic category, in [66], [50], [49]. Our present contribution is to consider essential singularities.…”

Section: E(d)mentioning

confidence: 99%

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Dynamics of singular complex analytic vector fields with essential singularities I

Alvarez-Parrilla¹,

Muciño–Raymundo²

2017

Conform. Geom. Dyn.

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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 ω X (with ω X (X) ≡ 1), orientable quadratic differentials ω X ⊗ ω X , global distinguished parameters Ψ X (z) = z ω X , and the Riemann surfaces 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 Ψ X .We show that under certain analytical conditions on Ψ 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 E , stemming from X(z) = e z ∂ ∂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 Re (X) 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 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 E(d) = {X(z) = λe P (z) ∂ ∂z } on the Riemann sphere, where P (z) is a polynomial of degree d and λ ∈ 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 ≥ 3 there are an infinite number of topological (phase portrait) classes of Re(X), for X ∈ E(d). These results are based on the work of R. Nevanlinna, A. Speisser and M. Taniguchi on entire functions Ψ 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 C vector fields X, with an essential singularity at ∞ ∈ C, whose Ψ −1 X have d logarithmic branch points over d finite asymptotic values and d logarithmic branch points over ∞.

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“…Summing up we get the following result (first introduced for (1)(2)(3)(4) in [50], [49], for (5) in the rational case [17], [18] and now expanded to cover (6) …”

Section: Definition 24 1 a Global Analytic Function ξ Is A Collectmentioning

confidence: 93%

Section: E(d)mentioning

confidence: 99%

“…Partial versions of the correspondence (1)-(6): J. Muciño-Raymundo et al [50], J. Muciño-Raymundo [49], A. Bustinduy et al [17], [18].…”

Section: E(d)mentioning

confidence: 99%

“…Flat metrics: K. Strebel [66], R. Peretz [57], J. Muciño-Raymundo et al [50], H. Masur et al [48], J. Muciño-Raymundo [49], M. Kontsevich et al [44], J. P. Bowman et al [15].…”

Section: E(d)mentioning

confidence: 99%

Section: E(d)mentioning

confidence: 99%

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Dynamics of singular complex analytic vector fields with essential singularities I

Alvarez-Parrilla¹,

Muciño–Raymundo²

2017

Conform. Geom. Dyn.

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On the problem of deciding whether a holomorphic vector field is complete

López

Muciño–Raymundo²

2000

Complex Analysis and Related Topics

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Pole dynamics and an integral of motion for multiple SLE(0)

Alberts,

Byun,

Kang

et al. 2024

Sel. Math. New Ser.

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We describe the Loewner chains of the real locus of a class of real rational functions whose critical points are on the real line. Our main result is that the poles of the rational function lead to explicit formulas for the dynamical system that governs the driving functions. Our formulas give a simple method for mapping the class of rational functions into solutions to a non-trivial system of quadratic equations, and for directly showing that the curves in the real locus satisfy geometric commutation. These results are entirely self-contained and have no reliance on probabilistic objects, but make use of an integral of motion for the Loewner chain that is motivated by ideas from conformal field theory. We also show that the dynamics of the driving functions are a special case of the Calogero–Moser integrable system, restricted to a particular submanifold of phase space carved out by the Lax matrix. Our approach complements a recent result of Peltola and Wang, who showed that the real locus is the deterministic $$\kappa \rightarrow 0$$ κ → 0 limit of the multiple SLE$$(\kappa )$$ ( κ ) curves.

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Bifurcations of meromorphic vector fields on the Riemann sphere

Cited by 20 publications

References 8 publications

Dynamics of singular complex analytic vector fields with essential singularities I

Dynamics of singular complex analytic vector fields with essential singularities I

On the problem of deciding whether a holomorphic vector field is complete

Pole dynamics and an integral of motion for multiple SLE(0)

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