Liouville domains from Okounkov bodies

Given a strictly concave rational PL function [Formula: see text] on a complete [Formula: see text]-dimensional fan [Formula: see text], we construct an exact symplectic structure of finite volume on [Formula: see text] and a family of functions [Formula: see text] called polyhedral Hamiltonians. We prove that for each [Formula: see text] the one-periodic orbits of [Formula: see text] come in families corresponding to finitely many primitive lattice points of [Formula: see text] and determine their topology. When [Formula: see text] is negative on the rays of [Formula: see text], we show that the level sets of polyhedral Hamiltonians are hypersurfaces of contact type. As a byproduct, this construction provides a dynamical model for the singularities of toric varieties obtained as degenerations of Fano manifolds in any dimension via Okounkov bodies.