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

Linear difference equations, frieze patterns and combinatorial Gale transform