Non-stationary versions of fixed-point theory, with applications to fractals and subdivision

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 investigates “trajectories of maps defined by function systems” which are considered as generalizations of the traditional IFS. The significance of ’forward’ and ’backward’ trajectories of general sequences of maps is studied. The convergence properties of these trajectories constitute a non-stationary version of the classical fixed-point theory. Unlike the ordinary fractals which are self-similar at different scales, the attractors of trajectories of maps defined by function systems may have different structures at different scales.