Convex dynamics and applications

Abstract

This paper proves a theorem about bounding orbits of a time dependent dynamical system. The maps that are involved are examples in convex dynamics, by which we mean the dynamics of piecewise isometries where the pieces are convex. The theorem came to the attention of the authors in connection with the problem of digital halftoning. Digital halftoning is a family of printing technologies for getting full-color images from only a few different colors deposited at dots all of the same size. The simplest version consists in obtaining gray-scale images from only black and white dots. A corollary of the theorem is that for error diffusion, one of the methods of digital halftoning, averages of colors of the printed dots converge to averages of the colors taken from the same dots of the actual images. Digital printing is a special case of a much wider class of scheduling problems to which the theorem applies. Convex dynamics has roots in classical areas of mathematics such as symbolic dynamics, Diophantine approximation, and the theory of uniform distributions.