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Constant-to-one and onto global maps of homomorphisms between strongly connected graphs

Published online by Cambridge University Press:  19 September 2008

Masakazu Nasu
Masakazu Nasu
Affiliation:
Faculty of Engineering, Mie University, Tsu 514, Japan
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Abstract

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The global maps of homomorphisms of directed graphs are very closely related to homomorphisms of a class of symbolic dynamical systems called subshifts of finite type. In this paper, we introduce the concepts of ‘induced regular homomorphism’ and ‘induced backward regular homomorphism’ which are associated with every homomorphism between strongly connected graphs whose global map is finite-to-one and onto, and using them we study the structure of constant-to-one and onto global maps of homorphisms between strongly connected graphs and that of constant-to-one and onto homomorphisms of irreducible subshifts of finite type. We determine constructively, up to topological conjugacy, the subshifts of finite type which are constant-to-one extensions of a given irreducible subshift of finite type. We give an invariant for constant-to-one and onto homomorphisms of irreducible subshifts of finite type.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 3 , Issue 3 , September 1983 , pp. 387 - 413
Copyright
Copyright © Cambridge University Press 1983

References

REFERENCES

[1]Adler, R. L. & Marcus, B.. Topological entropy and equivalence of dynamical systems. Memoirs Amer. Math. Soc. 219 (1979).Google Scholar
[2]Berge, C.. Graphs and Hypergraphs. North-Holland: Amsterdam, 1973.Google Scholar
[3]Coven, E. M. & Paul, M. E.. Endomorphisms of irreducible subshifts of finite type. Math. Systems Theory 8 (1974), 167–175.CrossRefGoogle Scholar
[4]Coven, E. M. & Paul, M. E.. Sofic systems. Israel J Math. 20 (1975), 165–177.CrossRefGoogle Scholar
[5]Coven, E. M. & Paul, M. E.. Finite procedures for sofic systems. Monatsh. Math. 83 (1977), 265–278.Google Scholar
[6]Gantmacher, F. R.. The Theory of Matrices, Vols. I and II. Chelsea: New York, 1959.Google Scholar
[7]Harary, F.. Graph Theory. Addison-Wesley: Reading, Mass., 1969.CrossRefGoogle Scholar
[8]Hedlund, G. A.. Endomorphisms and automorphisms of the shift dynamical system. Math. Systems Theory 3 (1969), 320–375.CrossRefGoogle Scholar
[9]Jacobson, N.. Lectures in Abstract Algebra, Vol. II. Springer-Verlag: New York, 1953.CrossRefGoogle Scholar
[10]Kitchens, B.. An invariant for continuous factors of Markov shifts. Proc. Amer. Math. Soc. 83 (1981), 825–828.CrossRefGoogle Scholar
[11]Kitchens, B.. Continuity properties of factor maps in ergodic theory. Ph.D. Thesis, University of North Carolina, Chapel Hill (1981).Google Scholar
[12]Klein, B. G.. Homomorphisms of symbolic dynamical systems. Math. Systems Theory 6 (1972), 107–122.CrossRefGoogle Scholar
[13]Marcus, B.. Factors and extensions of full shifts. Monatsh. Math. 88 (1979), 239–247.CrossRefGoogle Scholar
[14]Nasu, M.. Local maps inducing surjective global maps of one-dimensional tessellation automata. Math. Systems Theory 11 (1978), 327–351.CrossRefGoogle Scholar
[15]Nasu, M.. Indecomposable local maps of tessellation automata. Math. Systems Theory 13 (1979), 81–93.CrossRefGoogle Scholar
[16]Nasu, M.. An interconnection of local maps inducing onto global maps. Discrete Applied Math. 2 (1980), 125–150.CrossRefGoogle Scholar
[17]Nasu, M.. Uniformly finite-to-one and onto extensions of homomorphisms between strongly connected graphs. Discrete Math. 39 (1982), 171–197.CrossRefGoogle Scholar
[18]Parry, W.. A finitary classification of topological Markov chains and sofic systems. Bull. London Math. Soc. 9 (1977), 86–92.CrossRefGoogle Scholar
[19]Parry, W. & Williams, R. F.. Block coding and a zeta function for finite Markov chains. Proc. London Math. Soc. (3) 35 (1977), 483–495.CrossRefGoogle Scholar
[20]Perles, M., Rabin, M. O., & Shamir, E.. The theory of definite automata. IEEE Trans. Electr. Comp. EC–12 (1963), 233–243.CrossRefGoogle Scholar
[21]Williams, R. F.. Classification of subshifts of finite type. Ann. of Math. 98 (1973), 120–153; Errata: Ann. of Math. 99 (1974), 380–381.CrossRefGoogle Scholar