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The zeta functions of renewal systems

Published online by Cambridge University Press:  26 March 2010

SOONJO HONG
SOONJO HONG*
Affiliation:
Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea (email: [email protected])
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Abstract

A renewal system is a shift space consisting of all bi-infinite free concatenations of some finitely many words. It is known that the zeta function of a renewal system generated by a circular set coincides with the generating function of the circular set. We extend this result to the systems generated by almost circular sets. Also we establish the formulas for the zeta functions of the systems generated by codes and pure codes, respectively. Using the formulas, we show that pure codes generating shifts of finite type are circular.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 3 , June 2011 , pp. 875 - 890
Copyright
Copyright © Cambridge University Press 2010

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References

[1]Béal, M. P.. Puissance extérieure d’un automate déterministe, application au calcul de la fonction zêta d’un système sofique. RAIRO Inform. Théor. Appl. 29 (1995), 85–103.CrossRefGoogle Scholar
[2]Berstel, J. and Perrin, D.. Theory of Codes. Academic Press, New York, 1985.Google Scholar
[3]Bowen, R.. On Axiom A Diffeomorphisms (AMS-CBMS Reg. Conf., 35). American Mathematical Society, Providence, RI, 1978.Google Scholar
[4]Bowen, R. and Lanford, O. E.. Zeta functions of restrictions of the shift transformation. Proc. Symp. Pure Math. A.M.S. 14 (1970), 43–50.CrossRefGoogle Scholar
[5]Boyle, M.. A zeta function for homomorphisms of dynamical systems. J. Lond. Math. Soc. 40 (1989), 355–368.CrossRefGoogle Scholar
[6]Coven, E. and Paul, M.. Endomorphisms of irreducible shifts of finite type. Math. Syst. Theory 8 (1974), 167–175.CrossRefGoogle Scholar
[7]Foata, D.. A combinatorial proof of Jacobi’s identity. Ann. Discrete Math. 6 (1981), 125–135.CrossRefGoogle Scholar
[8]Fried, D.. Finitely presented dynamical systems. Ergod. Th. & Dynam. Sys. 7 (1987), 489–507.CrossRefGoogle Scholar
[9]Hong, S. and Shin, S.. The entropies and periods of renewal systems. Israel J. Math. 172 (2009), 9–27.CrossRefGoogle Scholar
[10]Keller, G.. Circular codes, loop counting, and zeta-functions. J. Combin. Theory Ser. A 66 (1991), 75–83.CrossRefGoogle Scholar
[11]Kim, K. H., Ormes, N. S. and Roush, F. W.. The spectra of nonnegative integer matrices via formal power series. J. Amer. Math. Soc. 12 (2000), 773–806.CrossRefGoogle Scholar
[12]Lavallée, S. and Reutenauer, C.. On a zeta function associated with automata and codes. Theoret. Comput. Sci. 381 (2007), 266–273.CrossRefGoogle Scholar
[13]Lind, D. and Marcus, B.. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[14]Manning, A.. Axiom A diffeomorphisms have rational zeta functions. Bull. Lond. Math. Soc. 3 (1971), 215–220.CrossRefGoogle Scholar
[15]Parry, W. and Tuncel, S.. On the stochastic and topological structure of Markov chains. Bull. Lond. Math. Soc. 14 (1982), 16–27.CrossRefGoogle Scholar
[16]Silva, P. V.. A note on pure and p-pure languages. Acta Inform. 39 (2003), 579–595.CrossRefGoogle Scholar
[17]Stanley, R. P.. Enumerative Combinatorics, Vol. 1. Wadsworth and Brooks/Cole, Monterey, CA, 1986.CrossRefGoogle Scholar
[18]Williams, S.. Notes on renewal systems. Proc. Amer. Math. Soc. 110 (1990), 851–853.CrossRefGoogle Scholar