FIXED POINTS FEATURES IN N-POINT GRAVITATIONAL LENSES

A set of fixed points in N-point gravitational lenses is studied in the paper. We use complex form of lens mapping to study fixed points. There are some merits of using a complex form over coordinate. In coordinate form gravitational lens is described by a system of two equations and in complex form is described by one equation. We transform complex equation of N-point gravitational lens into polynomial equation. It is convenient to study polynomial equation. Lens mapping presented as a linear combination of two mappings: complex analytical and identity. Analytical mapping is specified by deflection function. Fixed points are roots of deflection function. We show, that all fixed points of lens mapping appertain to the minimal convex polygon. Vertices of the polygon are points into which dimensionless point masses are. Method of construction of fixed points in N-point gravitational lens is shown. There are no fixed points in 1-point gravitational lens. We study properties of fixed points and their relation to the center of mass of the system. We obtained dependence of distribution of fixed points on center of mass. We analyzed different possibilities of distribution in N-point gravitational lens. Some cases, when fixed points merge with the center of mass are shown. We show a linear dependence of fixed point on center of mass in 2-point gravitational lens and we have built a model of this dependence. We obtained dependence of fixed point to center of mass in 3-point lens in case when masses form a triangle or line. In case of triangle, there are examples when fixed points merges. We study conditions, when there are no one-valued dependence of distribution of fixed points in case of 3-points gravitational lens and more complicated lens.

Effect of gravitational lensing has evolved from a method of verification of general relativity to separate field over the years. Using gravitational lenses, stars, stars system and even planets can be found.
Often numerical methods were used to analyze gravitational lenses and analytically were described only special symmetrical systems [1,2]. But these days, gravitational lenses are studied using analytical methods, particularly using methods of algebraic geometry [3][4][5][6]. With algebraic geometry methods we were able to study an Einstein ring -an important object in gravitational lensing -and prove that it is the only extended image in N-point gravitational lenses [4].
Circle source in N-point gravitational lens of any configurations was built using methods of algebraic geometry [7]. It is important to study fixed points in the theory of gravitational lenses [8]. Fixed points were researched before [9]. In paper [9] were shown examples of fixed points.
This paper proposes methods to study the distribution of fixed points. We show the distribution of fixed points and study its properties in N-point gravitational lenses.
The goal of this work is studying of a set of fixed points in N-point gravitational lenses using analytical methods, finding features of fixed points.

FORMULATION OF THE PROBLEM
N-point gravitational lens can be described by following equation where n m -are dimensionless masses whose position in the plane of the lens is determined by normalized radius-vectors

EEJP. 3 (2019) Volodymyr Shablenko
Unification of source plane and image plane is referred as picture plane. Mapping (2) can be described by system of equations:

LENS MAPPING COMPLEXIFICATION
It is more convenient to use complex form of lens mapping in order to find fixed points [8]. We introduce new complex variables: Using new variables, mapping can be described by following equation: And function is complex conjugated to ω and defined: It is convenient to use w and ω to find fixed points of lens mapping, except that it is more convenient to use w .
We use properties of complex mapping that were studied in [8]. It is known [8,11] that deflection function ( ) w w z = is analytical. There are some theorems.

N-POINT LENSES
Consider N-point gravitational lens. We change masses of point lenses and their positions are set to constant. Hence, fixed points depend on N parameters (masses of lenses). Besides, point masses are bounded by masses normalization equation. There is a necessary condition to find the parameters of fixed points distribution parameters. We need the same number of free parameters as number of equations. Therefore, we need either to decrease the number of parameters or to increase the number of equations.
To decrease number of equations we can express masses in terms of masses center of point masses system. It is convenient to describe 2-or 3-points gravitational lenses accordingly to this method. But we need to introduce new additional conditions to describe more complicated systems.
Deflection function of 1-point lens is: There are no fixed points in 1-point gravitational lens.
Deflection function of 2-points lens is: Fixed point is given by: We can also use equation for center of mass: If we express fixed point using the center of mass, we obtain:

EEJP. 3 (2019) Volodymyr Shablenko
We can set different parameterization for point masses in this case. There are some different mass parameterizations in 3-point gravitational lenses on ( fig.1-4). There are gravitational lenses with masses that form triangle on ( fig.1-2) and line in ( fig. 3-4).
We can describe a fixed point using the center of mass: If point masses form a triangle, we can obtain one-valued dependence of fixed points on center of mass. There is only 1 pair of fixed points per each center of mass. We put point masses in positions with 1 We obtain equation on fixed points:

z A A A A A A x A y A A A A A A A x A y A A A A x y A A A A A A
Using technical computing system, Wolfram Mathematica, we built mathematical model that shows the distribution of fixed points. A parameter in the model is the center of mass. To build a more general picture, we set simultaneously horizontal line of the possible center of mass. Thus, we can see 1 line of the center of mass and 2 curves of fixed points that match centers of masses.
Using our model, we obtain when 3-point lens forms equilateral triangle we have 2 fixed points that merge (Fig.5-6). In case when the center of mass of the triangle is geometrical center of the triangle, fixed points are located in 1 point. Situation, when masses are almost equal is shown on (Fig. 5) and equal masses are shown on (Fig.6). Consider scalene triangle. There was an assumption in [8] that fixed points can merge only in case when point masses form regular polygon. But in case of scalene triangle we have found cases of merging of fixed points. There is a configuration of 3-point gravitational lens when in a forms scalene triangle with the merging of fixed points on ( Fig.7-8). A mass configuration with two separate fixed points is shown in (Fig. 7) and configuration with merged fixed points is shown on (Fig. 8). If point masses are located along line, number of equations, that are used to describe masses, decreases by 1. In this case we need to introduce 1 additional parameter. The same center of mass can describe different distribution of masses. Considering this feature, we can obtain different fixed points for the same center of mass. We introduce additional parameter, for example,  (Fig.10-12). Triangle consists of 4 masses is shown on (Fig.  10), the square is shown in (Fig. 11) and triangle with 1 internal mass is shown in (Fig. 12).
We need to introduce more additional parameters to study gravitational lenses with 3 or more-point masses. Thus, it is more complicated to apply this method and it is only convenient to study 2-and 3-point gravitational lenses.