  In this paper we study the problem of the existence of a least-action
principle for invertible, second-order dynamical systems, discrete in time and
space. We show that, when the configuration space is finite, a least-action
principle does not exist for such systems. We dichotomize discrete dynamical
systems with infinite configuration spaces into those of finite type for which
this theorem continues to hold, and those not of finite type for which it is
possible to construct a least-action principle. We also show how to recover an
action by restriction of the phase space of certain second-order discrete
dynamical systems. We provide numerous examples to illustrate each of these
results.
