Exact numerical studies of Hamiltonian maps: Iterating without roundoff error

For many important Hamiltonian maps (e.g., the standard map) it is possible to construct related mappings that (i) carry a lattice into itself; (ii) approach the original map as the lattice spacing is decreased; (iii) can be iterated exactly using integer arithmetic; and (iv) are Hamiltonian themselves. We compare these lattice maps to maps that use floating-point arithmetic to evaluate the original map. We discuss the problems associated with roundoff error and we argue that lattice maps are superior to floating-point maps for the study of the long-term behaviour of Hamiltonian dynamical systems.