  We investigate the convergence towards periodic orbits in discrete dynamical
systems. We examine the probability that a randomly chosen point converges to a
particular neighborhood of a periodic orbit in a fixed number of iterations,
and we use linearized equations to examine the evolution near that
neighborhood. The underlying idea is that points of stable periodic orbit are
associated with intervals. We state and prove a theorem that details what
regions of phase space are mapped into these intervals (once they are known)
and how many iterations are required to get there. We also construct algorithms
that allow our theoretical results to be implemented successfully in practice.
