Transition phenomena for the attractor of an iterated function system*

Iterated function systems (IFSs) and their attractors have been central to the theory of fractal geometry almost from its inception. And contractivity functions in IFS has iterated systems. If are contractions, then is guaranteed a unique attractor. Recently, however, there an interest what occurs attractor at boundary between contractvity expansion IFS. That subject this paper. For family $F_t$ IFSs depending on real parameter $t>0$, existence properties two types transition attractors, called lower $A_{\bullet}$ upper $A^{\bullet}$, investigated. A main theorem states that, for wide class families, threshold $t_0$ such that $A_t$ $tt_0$. At $t_0$, $F_{t_0}$-invariant set $A^{\bullet}$ $A^{\bullet} = \lim_{t\rightarrow t_0} A_t$.