Ergodic Theory and Dynamical Systems

Table of Contents - 2005 - Volume 25, Issue 02  


Convex dynamics and applications

R. L. ADLER a1, B. KITCHENS a2, M. MARTENS a3, C. PUGH a4, M. SHUB a1p1 and C. TRESSER a1
a1 IBM, TJ Watson Research Center, Yorktown Heights, NY 10598-0218, USA (e-mail: rla@us.ibm.com, shub@math.toronto.edu, tresser@us.ibm.com)
a2 Mathematical Sciences Department, IUPUI-LD270, Indianapolis, IN 46202, USA (e-mail: bkitchens@math.iupui.edu)
a3 University of Groningen, Department of Mathematics, PO Box 800, 9700 AV Groningen, The Netherlands (e-mail: marco@math.rug.nl)
a4 Mathematics Department, University of California, Berkeley, CA 94720, USA (e-mail: pugh@math.berkeley.edu)

Article author query
adler rl   [Google Scholar] 
kitchens b   [Google Scholar] 
martens m   [Google Scholar] 
pugh c   [Google Scholar] 
shub m   [Google Scholar] 
tresser c   [Google Scholar] 
 

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.

(Received March 4 2004)
(Revised June 10 2004)


Correspondence:
p1 Department of Mathematics, University of Toronto, 100 St. George Street, Toronto, Ontario M5S 3G3, Canada