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Open index pairs, the fixed point index and rationality of zeta functions

Published online by Cambridge University Press:  19 September 2008

Marian Mrozek
Marian Mrozek
Affiliation:
Katedra Informatyki, Uniwersytet Jagielloński, ul. Kopernika 27, 31–501 Kraków, Poland
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Abstract

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We define open index pairs of an isolated invariant set, prove their existence and compute the fixed point index of an isolating neighbourhood in terms of the Lefschetz number of a certain map associated with the open index pair. We use this to establish rationality of zeta functions and Lefschetz zeta functions.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 10 , Issue 3 , September 1990 , pp. 555 - 564
Copyright
Copyright © Cambridge University Press 1990

References

REFERENCES

[1]Artin, E. & Mazur, B.. On periodic points. Ann. Math. (2) 81 (1965), 82–99.CrossRefGoogle Scholar
[2]Conley, C. C.. Isolated invariant sets and the Morse index. CBMS Regional Conf. Ser. Math. No. 38, Amer. Math. Soc.: Providence, R. I., 1978.Google Scholar
[3]Conley, C. C. & Easton, R. W.. Isolated invariant sets and isolating blocks. Trans. Amer. Math. Soc. 158 (1971), 35–61.Google Scholar
[4]Fenske, C. C. & Peitgen, H. O.. On fixed points of zero index in asymptotic fixed point theory. Pac. J. Math. 66 No. 2 (1976), 391–410.Google Scholar
[5]Fournier, G.. Généralisations du théorème de Lefschetz pour des espaces non-compacts I, II, III. Bull. Acad. Polon. Sci., Ser. Math. Astr. Ph. 23 (1975), 693–711.Google Scholar
[6]Franks, J.. Homology and dynamical systems. CBMS Reg. Conf. Series Math. No. 49, Amer. Math. Soc.: Providence, R. I. 1982.Google Scholar
[7]Fried, D.. Rationality for isolated expansive sets. Adv. Math. 65 (1987), 35–38.CrossRefGoogle Scholar
[8]Granas, A.. The Leray-Schauder index and the fixed point theory for arbitrary ANRs. Bull. Soc. Math. France 100 (1972), 209–228.Google Scholar
[9]Manning, A.. Axiom A diffeomorphisms have rational zeta functions. Bull. London Math. Soc. 3 (1971), 215–220.CrossRefGoogle Scholar
[10]Mrozek, M.. The fixed point index of a translation operator of a semiflow. Univ. lag. Adta Math. 27 (1988), 13–22.Google Scholar
[11]Mrozek, M.. Index pairs and the fixed point index for semidynamical systems with discrete time. Fund. Math. 133 (1989), 178–192.Google Scholar
[12]Mrozek, M.. Leray functor and the cohomological Conley index for discrete dynamical systems. Trans. Amer. Math. Soc. 318 (1) (1990), 149–178.CrossRefGoogle Scholar
[13]Robbin, J. W. & Salamon, D.. Dynamical systems, shape theory and the Conley index. Ergod. Th. and Dynam. Sys. 8* (1988), 375–393.Google Scholar
[14]Rybakowski, K.. On the homotopy index for infinite-dimensional semiflows. Trans. Amer. Math. Soc. 269 (2) (1982), 351–382.CrossRefGoogle Scholar
[15]Smale, S.. Differentiable dynamical systems. Bull. Am. Math. Soc. 73 (1967), 747–817.CrossRefGoogle Scholar
[16]Spanier, E.. Algebraic Topology. McGraw-Hill: New York, 1966.Google Scholar
[17]Wilson, F. W. Jr, Smoothing derivatives of functions and applications. Trans. Amer. Math. Soc. 139 (1969), 413–428.Google Scholar
[18]Wilson, F. W. Jr, & Yorke, J. A.. Lyapunov functions and isolating blocks. J. Diff. Eq. 13 (1973), 106–123.CrossRefGoogle Scholar