  Symplectic maps are routinely used to describe single-particle dynamics in
circular accelerators. In the case of a linear accelerator map, the rotation
number (the betatron frequency) can be easily calculated from the map itself.
In the case of a nonlinear map, the rotation number is normally obtained
numerically, by iterating the map for given initial conditions, or through a
normal form analysis, a type of a perturbation theory for maps. Integrable
maps, a subclass of symplectic maps, allow for an analytic evaluation of their
rotation numbers. In this paper we propose an analytic expression to determine
the rotation number for integrable symplectic maps of the plane and present
several examples, relevant to accelerators. These new results can be used to
analyze the topology of the accelerator Hamiltonians as well as to serve as the
starting point for a perturbation theory for maps.
