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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

##### MSC:
 55M25 Degree, winding number 58C25 Differentiable maps on manifolds
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##### References:
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