  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 expressions 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.
