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The pentagram map is a discrete dynamical system defined on the moduli spaceof polygons in the projective plane. This map has recently attracted aconsiderable interest, mostly because its connection to a number of differentdomains, such as: classical projective geometry, algebraic combinatorics,moduli spaces, cluster algebras and integrable systems. Integrability of thepentagram map was conjectured by R. Schwartz and later proved by V. Ovsienko,R. Schwartz and S. Tabachnikov for a larger space of twisted polygons. In thispaper, we prove the initial conjecture that the pentagram map is completelyintegrable on the moduli space of closed polygons. In the case of convexpolygons in the real projective plane, this result implies the existence of atoric foliation on the moduli space. The leaves of the foliation carry affinestructure and the dynamics of the pentagram map is quasi-periodic. Our proof isbased on an invariant Poisson structure on the space of twisted polygons. Weprove that the Hamiltonian vector fields corresponding to the monodoromyinvariants preserve the space of closed polygons and define an invariant affinestructure on the level surfaces of the monodromy invariants.

Liouville-Arnold integrability of the pentagram map on closed polygons