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.
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