Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-24T15:56:16.020Z Has data issue: false hasContentIssue false
  • English
  • Français

Embeddings of dynamical systems into cellular automata

Published online by Cambridge University Press:  01 February 2009

JOHANNES MÜLLER  and
CHRISTOPH SPANDL
JOHANNES MÜLLER
Affiliation:
Technical University Munich, Centre for Mathematical Sciences, Boltzmannstr. 3, D-85748 Garching/Munich, Germany (email: johannes.mueller@gsf.de)
CHRISTOPH SPANDL
Affiliation:
Institute for Theoretical Computer Science and Mathematics, University of the Federal Armed Forces Munich, D-85577 Neubiberg, Germany (email: christoph.spandl@unibw.de)
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that general topological dynamical systems on metric Cantor spaces can be embedded into cellular automata defined on Cayley graphs. Furthermore, we prove that it is possible to construct this embedding in such a way that topological entropy is preserved.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 1 , February 2009 , pp. 165 - 177
Copyright
Copyright © 2008 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470). Springer, Berlin, 1975.CrossRefGoogle Scholar
[2]Boyle, M., Fiebig, D. and Fiebig, U.. Residual entropy, conditional entropy and subshift covers. Forum Math. 14 (2002), 713–757.CrossRefGoogle Scholar
[3]Brin, M. and Stuck, G.. Introduction to Dynamical Systems. Cambridge University Press, Cambridge, 2002.CrossRefGoogle Scholar
[4]Ceccherini-Silberstein, T. G., Machì, A. and Scarabotti, F.. Amenable groups and cellular automata. Ann. Inst. Fourier 49 (1999), 673–685.CrossRefGoogle Scholar
[5]Demongeot, J., Goler, E. and Tchuente, M.. Dynamical Systems and Cellular Automata. Academic Press, Orlando, FL, 1985.Google Scholar
[6]Durand, B.. The surjectivity problem for 2D cellular automata. J. Comput. System Sci. 49 (1994), 718–724.CrossRefGoogle Scholar
[7]Hedlund, G. A.. Endomorphisms and automorphisms of the shift dynamical system. Math. Syst. Theory 3 (1969), 320–375.CrossRefGoogle Scholar
[8]Hurd, L., Kari, J. and Culik, K.. The topological entropy of cellular automata is uncomputable. Ergod. Th. & Dynam. Sys. 12 (1992), 255–265.CrossRefGoogle Scholar
[9]Kari, J.. Reversibility and surjectivity problems of cellular automata. J. Comput. System Sci. 48 (1994), 149–182.CrossRefGoogle Scholar
[10]Machì, A. and Mignosi, F.. Garden of Eden configurations for cellular automata on Cayley graphs of groups. SIAM J. Discrete Math. 6 (1993), 44–56.CrossRefGoogle Scholar
[11]Reddy, W. L.. Lifting expansive homeomorphisms to symbolic flows. Math. Syst. Theory 2 (1968), 91–92.CrossRefGoogle Scholar
[12]Róka, Z.. Simulations between cellular automata. Theoret. Comput. Sci. 225 (1999), 81–111.CrossRefGoogle Scholar
[13]Schupp, P. E.. Arrays, Automata and Groups – Some Interconnections (Automata Networks, Proceedings of the LITP Spring School on Theoretical Computer Sciences, Argelès-Villages, May 1986) (Lecture Notes in Computer Science, 316). Ed. C. Choffrut. Springer, Berlin, 1988.Google Scholar
[14]Walters, P.. An Introduction to Ergodic Theory. Springer, New York, 1982.CrossRefGoogle Scholar