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In this paper, we study the distribu- tion of images from a point source in N - point gravi- tational lenses. It is well known that in a Schwarzschild lens (N = 1) from a point source, there are always two images. Moreover, one of them is always inside the Einstein ring, and the second outside it. It follows that: a) the image plane is divided into two areas; b) in each area there is always only one image; c) the source plane, with the exception of the caustic point (origin), is uniquely mapped onto each of the two areas of the image plane. In our study, we describe an algorithm that allows us to determine the single-valued regions in a 2-point gravitational lens and demonstrate it using an exam- ple of a binary symmetric lens in which the distance between the point masses is 1. We have shown that in this case the full prototype of the caustic divides the image plane into eight simply connected areas that have the following properties: a) if the point source is inside the caustic, then it has five images in five (internal) areas; b) if the point source is outside the caustic, then it has three images in three (external) areas; c) in each area there can be no more than one image; d) if the image of a point source is located in one of the five internal areas, then the remaining four also have images, while none of the three external areas have images of the source; e) if the image of the source is located in one of the three external areas, then its images also exist in the remaining two, while none of the five internal areas contains a prototype of this source; f) a caustic is a continuous, piecewise smooth, closed Jordan curve that has a finite number of singular points; each smooth, open part of the caustic, the ends of which are singular points (the caustic arc) has four inverse images, of which only one belongs to the critical set, the caustic arcs are positively oriented (when going around, the interior of the caustic remains to the left; g) the boundaries of the regions consist of arcs (closure of the image of arcs) with a hereditary orien- tation; all eight regions are divided into the following two classes: four regions in which the orientation of the boundary coincides with the orientation on the caustic and four regions in which the orientation of the bound- ary is opposite to the orientation on the caustic.

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.

In this paper, we study the correlation between the number of solutions of a system of lens equations and the number of source images that a gravitational lens has. We defined the concept of an image in a gravitational lens. To find the set of solutions of a system of lens equations, we applied methods of algebraic geometry.