Thresholds for one-parameter families of affine iterated function systems*

Abstract

This paper examines thresholds for certain properties of the attractor of a general one-parameter affine family of iterated functions systems. As the parameter increases, the iterated function system becomes less contractive, and the attractor evolves. Thresholds are studied for the following properties: the existence of an attractor, the connectivity of the attractor, and the existence of non-empty interior of the attractor. Also discussed are transition phenomena between existence and non-existence of an attractor.

Transition Phenomena for the Attractor of an Iterated Function System

“…This paper is concerned with attractor phenomena at the transition between contractivity and expansion of a one-parameter IFS family, between the existence and non-existence of an attractor. To illustrate this kind of transition phenomena, consider the following family F t of IFSs that depends on a real parameter t > 0, which is based on [33,Example 1.1].…”

Contractive Iterated Function Systems Enriched with Nonexpansive Maps

“…Finally, in Sect. 7 we use presented machinery to prove that lower transition attractors considered recently by Vince [21] are attractors of enriched IFSs.…”

“…We start with giving some further denotations from [21]. Throughout this section, let F = {g, f 1 , .…”

“…. , f (N,t) }, t ≥ 0 is called in [21] as a bounded one-parameter similarity family (see [21, In the above framework, the set A F , * := {A ⊂ X : A is (F t0 , a * )-invariant} will be called the lower transition attractor of (F t ).…”

“…It can be wieved that the above presented proof can be exetended into wider setting. In forthcoming paper we are going to invesigate the existence of lower transition attractors (as well as upper transition attractors) from [21] for more general families F t .…”