  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 $A^{\bullet} =
\lim_{t\rightarrow t_0} A_t$.
