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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

55M25 Degree, winding number
58C25 Differentiable maps on manifolds
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