×

zbMATH — the first resource for mathematics

Fixed point indices of iterated maps. (English) Zbl 0583.55001
This article deals with some relations between the fixed point indices \(I(f^ n)\) \((n=1,2,\ldots)\) of the iterates \(f^ n\) of some map \(f: V\to Y\), where Y is a Euclidean neighbourhood retract, V is an open subset of Y. The author generalizes the formula \[ \sum_{\tau \subset P(n)}(-1)^{| \tau |} I(f^{n/\tau})\equiv 0 (mod n), \] (where P(n) is the set of all primes dividing n) to this case (the case when Y is a Banach space and f is compact was studied by the reviewer and M. A. Krasnosel’skij [Vestn. Yarosl. Univ. 12, 23-37 (1975)]). Moreover, the author describes some converse of this result, in particular the following: if s(n) is a sequence for which \[ \sum_{\tau \subset P(n)}(-1)^{| \tau |} s(n/\tau)\equiv 0 (mod n)\quad (n=2,3,\ldots) \] then there exists \(f: V\to Y\) for suitable V and Y such that \(I(f^ n)=s(n)\).
Reviewer: P.P.Zabrejko

MSC:
55M25 Degree, winding number
58C25 Differentiable maps on manifolds
PDF BibTeX Cite
Full Text: DOI EuDML
References:
[1] Bourbaki, N.: Groupes et algèbres de Lie. Chap. II. Paris: Hermann 1972 · Zbl 0244.22007
[2] Chow, S.-N., Mallet-Paret, J., Yorke, J.A.: A periodic orbit index which is a bifurcation invariant. IMPA proceedings on Geometric Dynamics. Lecture Notes in Mathematics, vol. 1007. Berlin-Heidelberg-New York: Springer 1983 · Zbl 0549.34045
[3] Dieck, T., tom: Transformation groups and representation theory. Lecture Notes in Mathematics, vol. 766. Berlin-Heidelberg-New York: Springer 1979 · Zbl 0445.57023
[4] Dold, A.: Lectures on algebraic topology. Berlin-Heidelberg-New York: Springer 1972 · Zbl 0234.55001
[5] Dold, A.: The fixed point index of fibre-preserving maps. Invent. Math.25, 281-297 (1974) · Zbl 0284.55007
[6] Dold, A.: The fixed point transfer of fibre-preserving maps. Math. Z.148, 215-244 (1976) · Zbl 0329.55007
[7] Guillemin, V., Pollack, A.: Differential topology. Eaglewood Cliff NJ: Prentice Hall 1974 · Zbl 0361.57001
[8] Knutson, D.: ?-rings and the representation theory of the symmetric group. Lecture Notes in Mathematics, vol. 308. Berlin-Heidelberg-New York: Springer 1973 · Zbl 0272.20008
[9] Okonek, Ch.: Bemerkung zurK-Theorie äquivarianter Endomorphismen. Archiv d. Math.40, 132-138 (1983) · Zbl 0592.18012
[10] Polya, G.: Über Potenzreihen mit ganzzahligen Koeffizienten. Math. Ann.77, 497-513 (1916) · JFM 46.0481.01
[11] Shub, M., Sullivan, D.: A remark on the Lefshetz fixed point formula for differentiable maps. Topology13, 189-191 (1974) · Zbl 0291.58014
[12] Steinlein, H.: Ein Satz über den Leray-Schauderschen Abbildungsgrad. Math. Z.126, 176-208 (1972) (MR 47 # 5667) · Zbl 0223.47023
[13] Zabreîko, P.P., Krasnosel’skii, M.A.: Iterations of operators, and fixed points. Dokl. Akad. Nauk SSSR196, 1006-1009 (1971); Voviet Math. Dokl.12, 1006-1009 (1971)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.