Advertisement

Acta Applicandae Mathematicae

, Volume 126, Issue 1, pp 245–252 | Cite as

Bi-resolving Graph Homomorphisms and Extensions of Bi-closing codes

  • Uijin Jung
  • In-Je Lee
Article

Abstract

Given two graphs G and H, there is a bi-resolving (or bi-covering) graph homomorphism from G to H if and only if their adjacency matrices satisfy certain matrix relations. We investigate the bi-covering extensions of bi-resolving homomorphisms and give several sufficient conditions for a bi-resolving homomorphism to have a bi-covering extension with an irreducible domain. Using these results, we prove that a bi-closing code between subshifts can be extended to an n-to-1 code between irreducible shifts of finite type for all large n.

Keywords

Covering Resolving Subamalgamation matrix Covering extension Bi-closing Shift of finite type 

Mathematics Subject Classification (2000)

37B10 05C50 05C70 37B40 

Notes

Acknowledgements

This paper was written during the authors’ graduate studies. We would like to thank our advisor Sujin Shin for her encouragement and good advice. We thank the referee for thorough suggestions which simplified and clarified many proofs, and Mike Boyle for comments which improved the exposition. The first named author was partially supported by TJ Park Postdoctoral Program. This work was also supported by the second stage of the Brain Korea 21 Project, The Development Project of Human Resources in Mathematics, KAIST in 2009.

References

  1. 1.
    Adler, R., Goodwyn, W., Weiss, B.: Equivalence of topological Markov shifts. Isr. J. Math. 27, 48–63 (1977) MathSciNetCrossRefGoogle Scholar
  2. 2.
    Adler, R., Marcus, B.: Topological entropy and equivalence of dynamical systems. Mem. Am. Math. Soc. 219 (1979) Google Scholar
  3. 3.
    Ashley, J.: An extension theorem for closing maps of shifts of finite type. Trans. Am. Math. Soc. 336, 389–420 (1993) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Ashley, J., Marcus, B., Perrin, D., Tuncel, S.: Surjective extensions of sliding-block codes. SIAM J. Discrete Math. 6, 582–611 (1993) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Boldi, P., Vigna, S.: Fibrations of graphs. Discrete Math. 243, 21–66 (2002) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Boyle, M.: Lower entropy factors of sofic systems. Ergod. Theory Dyn. Syst. 3, 541–557 (1983) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Boyle, M.: Open problems in symbolic dynamics. In: Burns, K., Dolgopyat, D., Pezin, Y. (eds.) Geometric and Probabilistic Structures in Dynamics. Contemporary Mathematics Series, vol. 469, pp. 69–118. American Mathematical Society, Providence (2008) CrossRefGoogle Scholar
  8. 8.
    Boyle, M., Krieger, W.: Almost Markov and shift equivalent sofic systems. In: Lecture Notes in Math., vol. 1342. Springer, Berlin (1988) Google Scholar
  9. 9.
    Boyle, M., Tuncel, S.: Infinite-to-one codes and Markov measures. Trans. Am. Math. Soc. 285, 657–684 (1984) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Deng, A., Sato, I., Wu, Y.: Homomorphisms, representations and characteristic polynomials of digraphs. Linear Algebra Appl. 423, 386–407 (2007) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Gross, J., Tucker, T.: Generating all graph coverings by permutation voltage assignments. Discrete Math. 18, 273–283 (1977) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Gross, J., Tucker, T.: Topological Graph Theory. Wiley-Interscience, New York (1987) MATHGoogle Scholar
  13. 13.
    Jung, U.: On the existence of open and bi-continuing codes. Trans. Am. Math. Soc. 363, 1399–1417 (2011) MATHCrossRefGoogle Scholar
  14. 14.
    Kim, K.H., Roush, F.: On the structure of inert automorphisms of subshifts. Pure Math. Appl., Ser. B 2, 3–22 (1991) MathSciNetMATHGoogle Scholar
  15. 15.
    Kim, K.H., Roush, F.: Free Z p actions on subshifts. Pure Math. Appl. 8, 293–322 (1997) MathSciNetMATHGoogle Scholar
  16. 16.
    Kitchens, B.: Symbolic Dynamics: One-Sided, Two-Sided and Countable State Markov Shifts. Springer, Berlin (1998) MATHGoogle Scholar
  17. 17.
    Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge (1995) MATHCrossRefGoogle Scholar
  18. 18.
    Nasu, M.: Constant-to-one and onto global maps of homomorphisms between strongly connected graphs. Ergod. Theory Dyn. Syst. 3, 387–413 (1983) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Trow, P.: Degrees of finite-to-one factor maps. Isr. J. Math. 71, 229–238 (1990) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    van Lint, J., Wilson, R.: A Course in Combinatorics, 2nd edn. Cambridge University Press, Cambridge (2001) MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of MathematicsAjou UniversitySuwonSouth Korea
  2. 2.The Jesuit Novitiate of St. Stanislaus in KoreaSuwonSouth Korea

Personalised recommendations