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 contractions for the same distance function. Here we consider IFSs composed of affine maps in the plane, and distance functions induced by elliptic norms the unit ball is an ellipse.. Every affine map of spectral radius less than 1 is contractive for some elliptic norm, but there exists no norm for which all such maps are contractive. Here we seek the set of all elliptic norms for which a given affine map is contractive the compatibility domain., and we show that the geometry of the compatibility domain depends on the nature of the eigenvalues: real and distinct, double or complex. An IFS will converge if and only if the compatibility domains have a nonempty intersection.