NON-STATIONARY EXTENSIONS OF BANACH FIXED-POINT THEOREM, WITH APPLICATIONS TO FRACTALS

Iterated Function Systems (IFS) 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 limits generated by subdivision schemes to
fractals generated by IFS. With an eye towards establishing connection between
non-stationary subdivision schemes and fractals, this talk introduces a non-stationary
extension of Banach fixed-point theorem. We introduce 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 is presented. Unlike the ordinary fractals which are self-similar
at different scales, the attractors of these trajectories may have different structures
at different scales. Joint work with Nira Dyn and Puthan Veedu Viswanathan.