A one-dimensional discrete-time system for which the transition map is state dependent is analyzed. For each transition a controller selects from among N transition maps according to a function of the current state. For the case in which the maps are contractions, sufficient conditions under which the system behavior can be modeled by a finite-state automaton are demonstrated. In this case the transient and steady-state behavior of the system can be computed. The feedback scheduling policy will have a periodic state, and the actual state of the system will converge exponentially to a periodic orbit. An example using a three-buffer switched server is given.