Dynamics of singular complex analytic vector fields with essential singularities II

Alvarez-Parrilla, Alvaro; Muciño–Raymundo, Jesús

doi:10.5427/jsing.2022.24a

Cited by 2 publications

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“…ii) X has a pole of multiplicity −k ≤ −1 at 0. iii) Up to local biholomorphism X is 1 z k ∂ ∂z . 5) i) Re (X) has any other topology different from (1)- (4).…”

Section: )mentioning

confidence: 99%

“…Proof. Here we provide a sketch of the proof, see [4] for full details. The associated single valued function is Ψ X (z), having 2d tracts at ∞.…”

Section: Holomorphic Families and Sporadic Examplesmentioning

confidence: 99%

“…The surgery idea first appeared in [40], moreover an approximation technique is developed in [4] §4.3. The pairing is equivalent to the fact that the phase portrait of the singular vector field Re (X) at ∞ ∈ C z has exactly 2d entire sectors E, see for instance [3] theorem 12.2.…”

Section: Holomorphic Families and Sporadic Examplesmentioning

confidence: 99%

“…The incomplete real trajectories of X (other than those arriving or exiting poles) coincide with the asymptotic trajectories of Ψ X . In previous works, we have exploited some of these ideas, see [31], [2] and [4].…”

Section: Introductionmentioning

confidence: 99%

“…Section 4 presents two subfamilies in the Speiser class. The first family is composed of functions of type Ψ X (z) = z P (ζ)e −E(ζ) dζ, with P (z), E(z) polynomials of degrees r and d ≥ 1, respectively; see Theorem 4.2 and our previous work [4]. The second family is given by functions of type Ψ X (z) = R • e 2πiz/T , where R(w) are rational functions of degree r ≥ 1, T ∈ C * .…”

Section: Introductionmentioning

confidence: 99%

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Geometry of transcendental singularities of complex analytic functions and vector fields

Alvarez-Parrilla¹,

Muciño–Raymundo²

2022

Preprint

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We study transcendental meromorphic functions with essential singularities on Riemann surfaces. Every function Ψ X has associated a complex vector field X. In the converse direction, vector fields X provide single valued or multivalued functions Ψ X . Our goal is to understand the relationship between the analytical properties of Ψ X , the singularities of its inverse Ψ −1 X and the geometric behavior of X. As first result, by examining the containment properties of the neighborhoods of the singularities of Ψ −1 X , we characterize when a singularity of Ψ −1 X over a singular value a, is either algebraic, logarithmic or non logarithmic. Secondly, to make precise the cooperative aspects between analysis and geometry, we present the systematic study of two holomorphic families of transcendental functions with an essential singularity at infinity, as well as some sporadic examples. As third stage, we study the incompleteness of the trajectories of the associated vector field X with essential singularities on a Riemann surface Mg of genus g. As an application, we provide conditions under which there exists an infinite number of (real) incomplete trajectories of X localized at the essential singularities. Furthermore, removing the incomplete trajectories decomposes the Riemann surface into real flow invariant canonical pieces. ContentsX 4. Holomorphic families and sporadic examples 4.1. Exponential families 4.2. Families of periodic vector fields 4.3. Sporadic examples 5. Incomplete trajectories 5.1. Existence of incomplete trajectories

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“…ii) X has a pole of multiplicity −k ≤ −1 at 0. iii) Up to local biholomorphism X is 1 z k ∂ ∂z . 5) i) Re (X) has any other topology different from (1)- (4).…”

Section: )mentioning

confidence: 99%

“…Proof. Here we provide a sketch of the proof, see [4] for full details. The associated single valued function is Ψ X (z), having 2d tracts at ∞.…”

Section: Holomorphic Families and Sporadic Examplesmentioning

confidence: 99%

Section: Holomorphic Families and Sporadic Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Geometry of transcendental singularities of complex analytic functions and vector fields

Alvarez-Parrilla¹,

Muciño–Raymundo²

2022

Preprint

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Geometry of transcendental singularities of complex analytic functions and vector fields

Alvarez-Parrilla,

Muciño-Raymundo

2024

Complex Manifolds

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On Riemann surfaces M M , there exists a canonical correspondence between a possibly multivalued function Ψ X {\Psi }_{X} whose differential is single-valued (i.e. an additively automorphic singular complex analytic function) and a vector field X X . From the point of view of vector fields, the singularities that we consider are zeros, poles, isolated essential singularities, and accumulation points of the above. The theory of singularities of the inverse function Ψ X ‒ 1 {\Psi }_{X}^{‒1} is extended from meromorphic functions to additively automorphic singular complex analytic functions. The main contribution is a complete characterization of when a singularity of Ψ X − 1 {\Psi }_{X}^{-1} is algebraic, is logarithmic, or arises from a zero with non-zero residue of X X . Relationships between analytical properties of Ψ X {\Psi }_{X} , singularities of Ψ X − 1 {\Psi }_{X}^{-1} and singularities of X X are presented. Families and sporadic examples showing the geometrical richness of vector fields on the neighbourhoods of the singularities of Ψ X − 1 {\Psi }_{X}^{-1} are studied. As applications, we have; a description of the maximal univalence regions for complex trajectory solutions of a vector field X X , a geometric characterization of the incomplete real trajectories of a vector field X X , and a description of the singularities of the vector field associated with the Riemann ξ {\rm{\xi }} -function.

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Dynamics of singular complex analytic vector fields with essential singularities II

Cited by 2 publications

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Geometry of transcendental singularities of complex analytic functions and vector fields

Geometry of transcendental singularities of complex analytic functions and vector fields

Geometry of transcendental singularities of complex analytic functions and vector fields

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