Transition Phenomena for the Attractor of an Iterated Function System

TL;DR

This paper investigates the transition phenomena of attractors in iterated function systems (IFSs) as they shift from contractive to expansive behavior, identifying threshold points where attractors emerge or vanish.

Contribution

It introduces the concepts of lower and upper transition attractors and establishes conditions under which these attractors appear at the boundary between contractivity and expansion.

Findings

Existence of a threshold parameter t0 for IFS attractors.

At t0, a unique invariant set A^{ullet} emerges as the limit of attractors.

For t > t0, the IFS has no attractor.

Abstract

Iterated function systems (IFSs) and their attractors have been central to the theory of fractal geometry almost from its inception. And contractivity of the functions in the IFS has been central to the theory of iterated functions systems. If the functions in the IFS are contractions, then the IFS is guaranteed to have a unique attractor. Recently, however, there has been an interest in what occurs to the attractor at the boundary between contractvity and expansion of the IFS. That is the subject of this paper. For a family $F_t$ of IFSs depending on a real parameter $t>0$, the existence and properties of two types of transition attractors, called the lower transition attractor $A_{\bullet}$ and the upper transition attractor $A^{\bullet}$, are investigated. A main theorem states that, for a wide class of IFS families, there is a threshold $t_0$ such that the IFS $F_t$ has a unique attractor $A_t$ for $t<t_0$ and no attractor for $t>t_0$. At the threshold $t_0$, there is an $F_{t_0}$-invariant set $A^{\bullet}$ such that $A^{\bullet} = \lim_{t\rightarrow t_0} A_t$.