Thresholds for one-parameter families of affine iterated function systems* *This work was partially supported by a grant from the Simons Foundation (322515 to Andrew Vince).

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.