Backward Asymptotics in S-Unimodal Maps

Abstract

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 [Formula: see text] was called by Hero the “special [Formula: see text]-limit” ([Formula: see text]-limit for short) of [Formula: see text]. In this article, we show that there is a hierarchy of [Formula: see text]-limits of points under iterations of a S-unimodal map: the size of the [Formula: see text]-limit of a point increases monotonically as the point gets closer and closer to the attractor. The [Formula: see text]-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.