Backward asymptotics in S-unimodal maps

TL;DR

This paper investigates the hierarchy of special alpha-limits in S-unimodal maps, revealing how the size of these limits grows as points approach the attractor, and characterizing their structure in relation to the non-wandering set.

Contribution

It introduces a hierarchy of sα-limits in S-unimodal maps and links their size to the proximity of points to the attractor, expanding understanding of backward dynamics.

Findings

The size of sα-limits increases monotonically towards the attractor.

The sα-limit of any point on the attractor is the entire non-wandering set.

The hierarchy reflects the structure of the S-unimodal map's graph.

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 $x$ was called by M.~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 non-wandering set. This hierarchy reflects the structure of the graph of a S-unimodal map recently introduced jointly by Jim Yorke and the present author.