Fixed points of mapping of N-point gravitational lenses - Kotvytskiy, A. T.; Shablenko, V. Yu.; Bronza, E. S. (Одеські астрономічні публікації, 2018)
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.