This paper develops a theory of dense orbital behavior in certain one-dimensional autonomous discrete-time systems. The systems in this class are all nonlinear and this paper is involved with a characterization of the state transitions such systems can have with respect to certain starting states. It is shown that (i) there is a relatively simple system in this class for which, for certain starting states, the set of state values assumed in time is a dense subset of the unit interval [0, 1], (ii) that this is the “only” system in this class exhibiting this density phenomenon, to within a natural type of system equivalence, and (iii) that necessary and sufficient conditions for determining if a system has dense orbits can be postulated: these conditions are related to properties of the set of points at which the transition function of the system, or one of its iterates, assumes the value of 0.
