Skip to main content
SpringerLink
Log in

The Myhill property for strongly irreducible subshifts over amenable groups

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

Let G be an amenable group and let A be a finite set. We prove that if X ⊂ A G is a strongly irreducible subshift then X has the Myhill property, that is, every pre-injective cellular automaton τ : X → X is surjective.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ceccherini-Silberstein T., Coornaert M.: The Garden of Eden theorem for linear cellular automata. Ergodic Theory Dyn. Syst. 26, 53–68 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ceccherini-Silberstein T., Coornaert M.: Induction and restriction of cellular automata. Ergodic Theory Dyn. Syst. 29, 371–380 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ceccherini-Silberstein T., Machì A., Scarabotti F.: Amenable groups and cellular automata. Ann. Inst. Fourier (Grenoble) 49, 673–685 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Fiorenzi F.: The Garden of Eden theorem for sofic shifts. Pure Math. Appl. 11, 471–484 (2000)

    MathSciNet  MATH  Google Scholar 

  5. Fiorenzi F.: Cellular automata and strongly irreducible shifts of finite type. Theoret. Comput. Sci. 299, 477–493 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Greenleaf, F.P.: Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, vol. 16, Van Nostrand Reinhold Co., New York (1969)

  7. Gromov M.: Endomorphisms of symbolic algebraic varieties. J. Eur. Math. Soc. (JEMS) 1, 109–197 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hedlund G.A.: Endomorphisms and automorphisms of the shift dynamical system. Math. Syst. Theory 3, 320–375 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kurka, P.: Topological and symbolic dynamics, Cours Spécialisés [Specialized Courses], vol. 11. Société Mathématique de France, Paris (2003)

  10. Lind D., Marcus B.: An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  11. Moore, E.F.: Machine models of self-reproduction, Proc. Symp. Appl. Math. vol. 14, pp. 17–34. American Mathematical Society, Providence (1963)

  12. Myhill J.: The converse of Moore’s Garden-of-Eden theorem. Proc. Am. Math. Soc. 14, 685–686 (1963)

    MathSciNet  MATH  Google Scholar 

  13. Paterson A.L.T.: Amenability, Mathematical Surveys and Monographs, vol. 29. American Mathematical Society, Providence, RI (1988)

    Google Scholar 

  14. Weiss, B.: Sofic groups and dynamical systems, Sankhyā Ser. A, 62 (2000), pp. 350–359. Ergodic theory and harmonic analysis (Mumbai, 1999)

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria, Università del Sannio, C.so Garibaldi 107, 82100, Benevento, Italy

    Tullio Ceccherini-Silberstein

  2. Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René-Descartes, 67000, Strasbourg, France

    Michel Coornaert

Authors
  1. Tullio Ceccherini-Silberstein
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michel Coornaert
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tullio Ceccherini-Silberstein.

Additional information

Communicated by Klaus Schmidt.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ceccherini-Silberstein, T., Coornaert, M. The Myhill property for strongly irreducible subshifts over amenable groups. Monatsh Math 165, 155–172 (2012). https://doi.org/10.1007/s00605-010-0256-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-010-0256-2

Keywords

Mathematics Subject Classification (2000)

Search

Navigation