  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.
