Invariant periodic manifolds of discrete maps

We study the stability properties of nonlinear discrete-time systems around an invariant periodic manifold. We construct a special set of local coordinates that highlight the tangential and transverse dynamics of the system. Such transverse coordinates play a key role in our analyses. One finds, for example, that an invariant periodic manifold of a discrete time system may contain both stable and unstable components. This phenomenon may occur in sample-and-hold implementations but has no parallel in continuous time.