Thresholds for One-Parameter Families of Affine Iterated Function Systems

TL;DR

This paper investigates how the properties of attractors in one-parameter affine iterated function systems change as the system becomes less contractive, identifying thresholds for existence, connectivity, and interior presence.

Contribution

It introduces a systematic analysis of thresholds for key properties of attractors in affine IFS families as parameters vary.

Findings

Identified parameter thresholds for attractor existence.

Determined conditions for attractor connectivity.

Analyzed transition phenomena between attractor states.

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.