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When the nearly periodic system is also Hamiltonian, Noether\u2019s theorem implies the existence of a corresponding adiabatic invariant. We develop a discrete-time analog of Kruskal\u2019s theory. Nearly periodic maps are defined as parameter-dependent diffeomorphisms that limit to rotations along a<jats:italic>U<\/jats:italic>(1)-action. When the limiting rotation is non-resonant, these maps admit formal<jats:italic>U<\/jats:italic>(1)-symmetries to all orders in perturbation theory. For Hamiltonian nearly periodic maps on exact presymplectic manifolds, we prove that the formal<jats:italic>U<\/jats:italic>(1)-symmetry gives rise to a discrete-time adiabatic invariant using a discrete-time extension of Noether\u2019s theorem. When the unperturbed<jats:italic>U<\/jats:italic>(1)-orbits are contractible, we also find a discrete-time adiabatic invariant for mappings that are merely presymplectic, rather than Hamiltonian. 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