  M. Kruskal showed that each continuous-time nearly-periodic dynamical system
admits a formal $U(1)$ symmetry, generated by the so-called roto-rate. When the
nearly-periodic system is also Hamiltonian, Noether's theorem implies the
existence of a corresponding adiabatic invariant. We develop a discrete-time
analoue of Kruskal's theory. Nearly-periodic maps are defined as
parameter-dependent diffeomorphisms that limit to rotations along a
$U(1)$-action. When the limiting rotation is non-resonant, these maps admit
formal $U(1)$ symmetries to all orders in perturbation theory. For Hamiltonian
nearly-periodic maps on exact presymplectic manifolds, we prove that the formal
$U(1)$ symmetry gives rise to a discrete-time adiabatic invariant using a
discrete-time extension of Noether's theorem. When the unperturbed
$U(1)$-orbits are contractible, we also find a discrete-time adiabatic
invariant for mappings that are merely presymplectic, rather than Hamiltonian.
As an application of the theory, we use it to develop a novel technique for
geometric integration of non-canonical Hamiltonian systems on exact symplectic
manifolds.
