Backward Asymptotics in S-Unimodal Maps

While the forward trajectory of a point in a discrete dynamical system is always unique, in general, a point can have infinitely many backward trajectories. The union of the limit points of all backward trajectories through x was called by Hero the "special alpha-limit" (s alpha-limit for short) of x. In this article, we show that there is a hierarchy of s alpha-limits of points under iterations of a S-unimodal map: the size of the s alpha-limit of a point increases monotonically as the point gets closer and closer to the attractor. The s alpha-limit of any point of the attractor is the whole nonwandering set. This hierarchy reflects the structure of the graph of a S-unimodal map recently introduced jointly by Jim Yorke and the present author.