Geometry of transcendental singularities of complex analytic functions and vector fields

On Riemann surfaces <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>M</m:mi> </m:math> M , there exists a canonical correspondence between a possibly multivalued function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="normal">Ψ</m:mi> </m:mrow> <m:mrow> <m:mi>X</m:mi> </m:mrow> </m:msub> </m:math> {Ψ }X whose differential is single-valued (i.e. an additively automorphic singular complex analytic function) and a vector field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>X</m:mi> </m:math> 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mi mathvariant="normal">Ψ</m:mi> </m:mrow> <m:mrow> <m:mi>X</m:mi> </m:mrow> <m:mrow> <m:mo>‒</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> </m:math> {Ψ }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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mi mathvariant="normal">Ψ</m:mi> </m:mrow> <m:mrow> <m:mi>X</m:mi> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> </m:math> {Ψ }X⁻¹ is algebraic, is logarithmic, or arises from a zero with non-zero residue of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>X</m:mi> </m:math> X . Relationships between analytical properties of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="normal">Ψ</m:mi> </m:mrow> <m:mrow> <m:mi>X</m:mi> </m:mrow> </m:msub> </m:math> {Ψ }X , singularities of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mi mathvariant="normal">Ψ</m:mi> </m:mrow> <m:mrow> <m:mi>X</m:mi> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> </m:math> {Ψ }X⁻¹ and singularities of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>X</m:mi> </m:math> X are presented. Families and sporadic examples showing the geometrical richness of vector fields on the neighbourhoods of the singularities of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mi mathvariant="normal">Ψ</m:mi> </m:mrow> <m:mrow> <m:mi>X</m:mi> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> </m:math> {Ψ }X⁻¹ are studied. As applications, we have; a description of the maximal univalence regions for complex trajectory solutions of a vector field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>X</m:mi> </m:math> X , a geometric characterization of the incomplete real trajectories of a vector field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">