The N-point gravitational lens as cover and his the profile cover

Abstract

The study of mathematical models of gravitational lenses are not direct observations. A special place in such studies is the visualization of the lens model. The image of the source and its images in the N-point gravitational lens, in the picture plane, visualizes the mathematical model - the algebraic equation of the lens. Recently, the number of studies of the equation of the N-point gravitational lens by algebraic methods has increased [6–8]. Such studies make it possible to consider the gravitational lens not only as an algebraic, but also as a topological object.
In the work, the equation of the N-point gravitational lens in the complex form is studied. A bundle above the source plane is assigned to it. We investigated one subfamily of lens equations. A critical set of equations of this subfamily is a closed Jordan curve. To the equations of this subfamily we put in correspondence not only a vector bundle, but also a covering.
A method for describing coverings is developed for equations whose caustic in the finite plane is a closed Jordan curve (Jordan caustic). A special case of such coverings is coverings for the equation of an N-point gravitational lens, the critical set of which is a closed Jordan curve. These equations, also, have Jordan caustics. The method is similar to the method for describing Riemann surfaces of algebraic functions, graphs ‒ profiles.
The algorithm for constructing coverings and the developed method for describing these coverings illustrates an example of a cover given by a rational non-analytic function of a complex variable The covering surface has not only a Jordan caustic, but also a second-order branch point at an infinitely distant point.
The methods of the theory of functions of a complex variable, algebraic geometry, algebraic topology and graph theory are used.