Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T06:00:14.944Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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