Fixed points of mapping of N-point gravitational lenses

Анотація

In this paper, we study fixed points of N-point gravitational lenses. We use complex form of lens mapping to study fixed points. Complex form
has an advantage over coordinate one because we can describe N-point gravitational lens by system of two equation in coordinate form and we can describe it by one equation in complex form. We can easily  transform the equation, which describe N-point gravitational lens, into polynomial equation that is convenient to use for our research. In our work, we present lens mapping as a linear combination of two mapping: complex analytical and identity mapping. Analytical mapping is specified by analytical function (deflection function). We studied necessary and sufficient conditions for the existence of deflection function and proved some theorems. Deflection function is analytical, rational, its zeroes are fixed points of lens mapping and their number is from 1 to N-1, poles of
deflection function are coordinates of point masses, all poles are simple, the residues at the poles are equal to the value of point masses. We used Gauss-Lucas theorem and proved that all fixed points of lens mapping are in the convex polygon. Vertices of the polygon consist of point masses. We proved theorem that can be used to find all fixed
point of lens mapping. On the basis of the above, we conclude that one-point gravitational lens has no fixed points, 2-point lens has only 1 fixed point, 3-point lens has 1 or 2 fixed points. Also we present expres-
sions to calculate fixed points in 2-point and 3-point gravitational lenses. We present some examples of parametrization of point masses and distribution of fixed points for this parametrization.