Self-Similar Functions Generated by Cellular Automata

The self-similarity properties of the functions (closed relations) associated with one- and two-sided cellular automata are studied. It turns out that these functions are generated by sequential machines, and their graphs are fractal sets generated by hierarchical iterated function systems. The Hausdorff dimensions of the graphs is one for one-sided cellular automata and two for two-sided automata.

[1]  Alejandro Maass,et al.  Représentation par automate de fonctions continues de tore , 1996 .

[2]  B. Voorhees.  Computational Analysis of One-Dimensional Cellular Automata , 1995 .

[3]  Ethan Akin,et al.  The general topology of dynamical systems , 1993 .

[4]  Heinz-Otto Peitgen,et al.  Global analysis of self-similarity features of cellular automata: selected examples , 1995 .

[5]  T. Kamae.  A characterization of self-affine functions , 1986 .

[6]  I. Cornfeld.  Ergodic theory / I.P. Cornfeld, S.V. Fomin, Ya.G. Sinai , 1982 .

[7]  Heinz-Otto Peitgen,et al.  Self-affine curves and sequential machines , 1996 .

[8]  Rune Kleveland,et al.  Mixing properties of one-dimensional cellular automata , 1997 .

[9]  C. Caramanis.  What is ergodic theory , 1963 .

[10]  G. A. Edgar.  Measure, Topology, and Fractal Geometry , 1990 .

[11]  J.-P. Allouche.  Finite Automata in 1-D and 2-D Physics , 1990 .

[12]  Petr Kurka,et al.  Topological and measure-theoretic properties of one-dimensional cellular automata , 1997 .

[13]  Michael Frame,et al.  The Canopy and Shortest Path in a Self-Contacting Fractal Tree , 1999 .

[14]  J. Lense.  Elementarmathematik vom höheren Standpunkt aus I , 1926 .

[15]  Stephen Wolfram,et al.  Cellular Automata And Complexity , 1994 .

[16]  K. Sigmund,et al.  Ergodic Theory on Compact Spaces , 1976 .

[17]  M. Shirvani,et al.  On ergodic one-dimensional cellular automata , 1991 .

[18]  György Targonski,et al.  Topics in Iteration Theory , 1981 .

[19]  Max H. Garzon,et al.  Models of massive parallelism: analysis of cellular automata and neural networks , 1995 .

[20]  R. Daniel Mauldin,et al.  Hausdorff dimension in graph directed constructions , 1988 .