  Iterated functions system (IFS) is defined by specifying a set of functions
in a classical phase space, which act randomly on an initial point. In an
analogous way, we define a quantum iterated functions system (QIFS), where
functions act randomly with prescribed probabilities in the Hilbert space. In a
more general setting a QIFS consists of completely positive maps acting in the
space of density operators. We present exemplary classical IFSs, the invariant
measure of which exhibits fractal structure, and study properties of the
corresponding QIFSs and their invariant states.
