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

Attractors of sequences of function systems and their relation to non-stationary subdivision