Elliptic norms and the convergence of iterated function systems in the plane

The convergence of Iterated Function Systems (IFSs) is guaranteed by Banach 's fixed point theorem, which requires that all functions in the IFS are co ntractions for the same distance function. Here we consider IFSs composed o f affine maps in the plane, and distance functions induced by elliptic norm s (the unit ball is an ellipse). Every affine map of spectral radius less t han 1 is contractive for some elliptic norm, but there exists no norm for w hich all such maps are contractive, Here we seek the set of all elliptic no rms for which a given affine map is contractive (the compatibility domain), and we show that the geometry of the compatibility domain depends on the n ature of the eigenvalues: real and distinct, double or complex. An IFS will converge if and only if the compatibility domains have a nonempty intersec tion, (C) 1999 Academic Press.