  Iterated Function Systems (IFSs) have been at the heart of fractal geometry
almost from its origin, and several generalizations for the notion of IFS have
been suggested. Subdivision schemes are widely used in computer graphics and
attempts have been made to link fractals generated by IFSs to limits generated
by subdivision schemes. With an eye towards establishing connection between
non-stationary subdivision schemes and fractals, this paper introduces the
notion of "trajectories of maps defined by function systems" which may be
considered as a new generalization of the traditional IFS. The significance and
the convergence properties of 'forward' and 'backward' trajectories are
studied. In contrast to the ordinary fractals which are self-similar at
different scales, the attractors of these trajectories may have different
structures at different scales.
