  We study the space of linear difference equations with periodic coefficients
and (anti)periodic solutions. We show that this space is isomorphic to the
space of tame frieze patterns and closely related to the moduli space of
configurations of points in the projective space. We define the notion of
combinatorial Gale transform which is a duality between periodic difference
equations of different orders. We describe periodic rational maps generalizing
the classical Gauss map.
