Skip to main content
SpringerLink
Log in

Some Dynamical Properties of One-Dimensional Cellular Automata

  • Chapter
Dynamics of Complex Interacting Systems

Part of the book series: Nonlinear Phenomena and Complex Systems ((NOPH,volume 2))

Abstract

In this note we review some results about the ergodic theory and topological dynamics of one-dimensional cellular automata. First, we describe those classifications of cellular automata with respect to their attractors and equicontinuous points. In section 4 we study onto cellular automata, in particular we give some results concerning to positively expansive cellular automata. Finally, we describe some symbolic dynamics of the limit sets of cellular automata.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ashley, J., An Extension Theorem for Closing Maps of Shifts of Finite Type, Transactions AMS 336, 389–420 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  2. Adler, R., B. Marcus, Topological Entropy and Equivalence of Dynamical Systems, Memoirs AMS 219 (1979).

    Google Scholar 

  3. Amoroso, S., Y.N. Patt, Decision Procedure for Surjectivity and Injectivity of Parallel Maps for Tessellation Structures, J. Comput. System Sci. 6, 448–464 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  4. Boyle, M., Lower Entropy Factors of Sofic Systems, Ergodic Theory and Dynamical Systems 4, 541–557 (1984).

    Article  MathSciNet  Google Scholar 

  5. Blanchard, F., Cellular Automata and Transducers: A Topological View, manuscript (1993).

    Google Scholar 

  6. Blanchard, F., B. Host, A. Maass, Représentation par Automate de Fonctions Continues du Tore, Journal de Teorie des Nombres de Bordeaux,to appear.

    Google Scholar 

  7. Blanchard, F., A. Maass, Dynamical Behaviour of Coven’s Aperiodic Cellular Automata, Theoretical Computer Science,to appear.

    Google Scholar 

  8. Boyle, M., B. Kitchens, B. Marcus, A Note on Minimal Covers for Sofic Systems, Proceedings AMS 95, 403–411 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  9. Blanchard, F., A. Maass, Dynamical Properties of Positively-Expansive Cellular Automata, submitted (1994).

    Google Scholar 

  10. Coven, E.M., Topological Entropy of Block Maps, Proceedings AMS 78, 590594 (1980).

    Google Scholar 

  11. Coven, E.M., M. Paul, Endomorphisms of Irreducible Shifts of Finite Type, Mathematical Systems Theory 8, 167–175 (1974).

    Article  MathSciNet  Google Scholar 

  12. Culik, K., J. Pachi, S. Yu, On the Limit Sets of Cellular Automata, SIAM J. Comput. 18, 831–842 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  13. Denker, M., C. Grillenberger, K. Sigmund, Ergodic Theory on Compact Spaces, Lecture Notes 527, Springer-Berlin (1975).

    Google Scholar 

  14. Fischer, R., Sofic Systems and Graphs, Monatshefte für Mathematik 80, 179186 (1975).

    Google Scholar 

  15. Goles, E., A. Maass, S. Martínez, On the Limit Set of some Universal Cellular Automata, Theoretical Computer Science 110, 53–78 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  16. Gilman, R.H., Notes on Cellular Automata, preprint (1988).

    Google Scholar 

  17. Gilman, R.H., Classes of Linear Automata, Ergodic Theory and Dynamical Systems 7, 105–118 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  18. Glasner, E., B. Weiss, Sensitive Dependence on Initial Conditions, Non Linearity 6, 1067–1075 (1993).

    MathSciNet  ADS  MATH  Google Scholar 

  19. Hedlund, G.A., Endomorphisms and Automorphisms of the Shift Dynamical System, Mathematical Systems Theory 3, 320–375 (1969).

    Article  MathSciNet  MATH  Google Scholar 

  20. Hurd, L., J. Kari, K. Culik, The Topological Entropy of Cellular Automata is Uncomputable, Ergodic Theory and Dynamical Systems 12, 255–265 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  21. Hurley, M., Attractors in Cellular Automata, Ergodic Theory and Dynamical Systems 10, 131–140 (1990).

    MathSciNet  MATH  Google Scholar 

  22. Hurley, M., Ergodic Aspects of Cellular Automata, Ergodic Theory and Dynamical Systems 10, 671–685 (1990).

    MathSciNet  Google Scholar 

  23. Hurley, M., Attractors in Restricted Cellular Automata, Proceedings AMS 115, 563–571 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  24. Hurd, L., Formal Language Characterizations of Cellular Automaton Limit Sets, Complex Systems 1, 69–80 (1987).

    MathSciNet  MATH  Google Scholar 

  25. Hurd, L., The Application of Formal Language Theory to the Dynamical Behaviour of Cellular Automata, a dissertation presented to the Faculty of Princeton University in candidacy for the degree of doctor of philosophy (1988).

    Google Scholar 

  26. Hurd, L., Recursive Cellular Automata Invariant Sets, Complex Systems 4, 119–129 (1990).

    MathSciNet  MATH  Google Scholar 

  27. Kari, J., Rice’s Theorem for the Limit Sets of Cellular Automata, Theoretical Computer Science,to appear.

    Google Scholar 

  28. Kari, J., Decision Problems Concerning Cellular Automata, thesis of the University of Turku (1990).

    Google Scholar 

  29. Krieger, W., On the Subsystems of Topological Markov Chains, Ergodic Theory and Dynamical Systems 2, 195–202 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  30. Mirka, P., Languages, Equicontinuity, and Attractors in Linear Cellular Au-tornata, preprint (1994).

    Google Scholar 

  31. Lind, D.A., Application of Ergodic Theory and Sofic Systems to Cellular Automata, Physica 10 D, 36–44 (1984).

    Google Scholar 

  32. Lind, D.A., Entropies of Automorphisms of a Topological Markov Shift, Proceedings AMS 99, 589–595 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  33. Maass, A., On Sofic Limit Sets of Cellular Automata, Ergodic Theory and Dynamical Systems,to appear.

    Google Scholar 

  34. Maass, A., Some Coded Systems that are not Unstable Limit Sets of CA, Cellular Automata and Cooperative Systems, NATO-ASI series, Kluwer Ac. Publ. 396, 433–449 (1993).

    MathSciNet  Google Scholar 

  35. Morita, K., M. Harao, Computation Universality of One Dimensional Reversible Cellular Automata, Transactions IEICE 72, 758–762 (1989).

    Google Scholar 

  36. Milnor, J., On the Entropy Geometry of Cellular Automata, Complex Systems 2, 357–386 (1988).

    MathSciNet  ADS  MATH  Google Scholar 

  37. Nasu, M., Local Maps Inducing Surjective Global Maps of One-Dimensional Tessellation Automata, Mathematical Systems Theory 11, 327–351 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  38. Nasu, M., Idecomposable Local Maps of Tessellation Automata, Mathematical Systems Theory 13, 81–93 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  39. Nasu, M., An Interconnection of Local Maps Inducing onto Global Maps, Discrete Applied Mathematics 2, 125–150 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  40. Nasu, M., Textile Systems for Endomorphisms and Automorphisms of the Shift, Memoirs AMS 546 (1995).

    Google Scholar 

  41. Parry, W., Intrinsic Markov Chains, Transactions AMS 112, 55–66 (1964).

    Article  MathSciNet  MATH  Google Scholar 

  42. Parry, W., Symbolic Dynamics and Transformations of the Unit Interval, Transactions AMS 122, 368–378 (1966).

    Article  MathSciNet  MATH  Google Scholar 

  43. Shereshevsky, M.A., Ergodic Properties of Certain Surjective Cellular Automata, Monatshefte für Mathematik 114, 305–316 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  44. Shereshevsky, M.A., Lyapunov Exponents for One-Dimensional Cellular Automata, Journal of Nonlinear Science 2, 1–8 (1992).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  45. Shereshevsky, M.A., Expansiveness, Entropy and Polynomial Growth for Groups Acting on Subshifts by Automorphisms, Indagationes Mathematicae 4, 203–210 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  46. Shirvani, M., T. Rogers, On Ergodic One-Dimensional Cellular Automata, Communications in Mathematical Physics 136, 599–605 (1991).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  47. Weiss, B., Subshifts of Finite Type and Sofic Systems, Monatshefte für Mathematik 77, 462–474 (1973).

    Article  MATH  Google Scholar 

  48. Wolfram, S., Computation Theory of Cellular Automata, Communications in Mathematical Physics 96, 15–57 (1984).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  49. Wolfram, S., Twenty Problems in the Theory of Cellular Automata, Physica Scripta 9, 170–172 (1985).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Ingeniería Matemática, Universidad de Chile F.C.F.M., Casilla 170-3 Correo 3, Santiago, Chile

    Alejandro Maass

Authors
  1. Alejandro Maass
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Departamento de Ingeniería Matemática, F.C.F.M., Universidad de Chile, Santiago, Chile

    Eric Goles  & Servet Martínez  & 

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Maass, A. (1996). Some Dynamical Properties of One-Dimensional Cellular Automata. In: Goles, E., Martínez, S. (eds) Dynamics of Complex Interacting Systems. Nonlinear Phenomena and Complex Systems, vol 2. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1323-8_2

Download citation

  • DOI: https://doi.org/10.1007/978-94-017-1323-8_2

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4734-2

  • Online ISBN: 978-94-017-1323-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation