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An invariant for bounded-to-one factor maps between transitive sofic subshifts

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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We define a ‘core-matrix’ of a transitive sofic subshift, which is unique up to similarity for each transitive sofic subshift. We prove that if there exists a bounded-to-one factor map from one transitive sofic subshift to another, the block of the Jordan form of a core-matrix of this second subshift with non-zero eigenvalues is a principal submatrix of the Jordan form of a core-matrix of the first. We also prove that the subshifts that are almost of finite type are ‘spectrally of finite type’.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 5 , Issue 1 , March 1985 , pp. 89 - 105
Copyright
Copyright © Cambridge University Press 1985

References

REFERENCES

[1]Adler, R. L. & Marcus, B.. Topological entropy and equivalence of dynamical systems. Mem. Amer. Math. Soc. 219 (1979).Google Scholar
[2]Boyle, M.. Topological orbit equivalence and factor maps in symbolic dynamics. Ph. D. Thesis, University of Washington, Seattle (1983).Google Scholar
[3]Boyle, M., Kitchens, B. & Marcus, B.. A note on minimal covers for sofic systems. Preprint.Google 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]Fischer, R.. Sofic systems and graphs, Monatsh. Math. 80 (1975), 179–186.CrossRefGoogle Scholar
[7]Fischer, R.. Graphs and symbolic dynamics. In Colloq. Math. Soc. János Bolyai 16, Topics in Information Theory. Keszthely: Hungary, 1975.Google Scholar
[8]Gantmacher, F. R.. The Theory of Matrices, Vol. II. Chelsea: New York, 1959.Google Scholar
[9]Harrison, M. A.. Introduction to Switching and Automata Theory. McGraw-Hill: New York, 1965.Google Scholar
[10]Hedlund, G. A.. Endomorphisms and automorphisms of the shift dynamical system. Math. Syst. Theory 3 (1969), 320–375.CrossRefGoogle Scholar
[11]Inagaki, Y., Fukumura, T. & Matuura, H.. Some aspects of linear space automata. Inform, and Contr. 20 (1972), 439–479.CrossRefGoogle Scholar
[12]Jacobson, N.. Lectures in Abstract Algebra, Vol. II, Springer-Verlag: New York, 1953.CrossRefGoogle Scholar
[13]Kitchens, B.. Continuity properties of factor maps in ergodic theory. Ph. D. Thesis, University of North Carolina, Chapel Hill (1981).Google Scholar
[14]Kitchens, B.. An invariant for continuous factors of Markov shifts. Proc. Amer. Math. Soc. 83 (1981), 825–828.CrossRefGoogle Scholar
[15]Krieger, W.. On sofic systems I. Preprint, Institut für Angewandte Mathematik der Universität Heidelberg.Google Scholar
[16]Marcus, B.. Sofic systems and encoding data. Preprint, Department of Mathematics, University of North Carolina.Google Scholar
[17]Matuura, H., Inagaki, Y. & Fukumura, T.. A generalization of automata and its analysis. Records natl. Convention, IECE, Japan A67–45 (1968). (In Japanese.)Google Scholar
[18]nasu, M.. Uniformly finite-to-one and onto extensions of homomorphisms between strongly connected graphs. Discrete Math. 39 (1982), 171–197.CrossRefGoogle Scholar
[19]nasu, M.. Constant-to-one and onto global maps of homomorphisms between strongly connected graphs. Ergod. Th. & Dynam. Sys. 3 (1983), 387–413.CrossRefGoogle Scholar
[20]Parry, W.. A finitary classification of topological Markov chains and sofic systems. Bull. London Math. Soc. 9 (1977), 86–92.CrossRefGoogle Scholar
[21]Paz, A.. Introduction to Probabilistic Automata. Academic Press: New York, 1971.Google Scholar
[22]Walters, P.. An Introduction to Ergodic Theory. Springer-Verlag: New York, 1982.CrossRefGoogle Scholar
[23]Weiss, B.. Subshifts of finite type and sofic systems. Monatsh. Math. 77 (1973), 462–474.CrossRefGoogle Scholar