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Fixed point indices of iterated smooth maps in arbitrary dimension. (English) Zbl 1247.37020

Let \(U\) be an open neighborhood of \(0\) in \(\mathbb R^m\) and let \(U\to \mathbb R^m\) be a \(C^1\)-map such that for each iterate of \(f\) the point \(0\) is an isolated fixed point. For \(n=1,2,\dots\) denote by \(\mathrm{ind}(f^n,0)\in \mathbb Z\) the fixed point index of \(f^n\). The article [S.-N. Chow, J. Mallet-Paret and J. A. Yorke, Lect. Notes Math. 1007, 109–131 (1983; Zbl 0549.34045)] states various conditions satisfied by the sequence \(\{\mathrm{ind}(f^n,0)\}\). The authors of that article also posed the question whether those conditions are the only restrictions imposed on sequences of fixed point indices of iterates of smooth maps. If \(m=1\) the problem is easy; in the dimensions \(2\) and \(3\) the problem was positively solved: in the case \(m=2\) by I. K. Babenko and S. A. Bogatyj [Math. USSR, Izv. 38, No. 1, 1–26 (1992); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 55, No. 1, 3–31 (1991; Zbl 0742.58027)], and in the case \(m=3\) by the first and the third author in [Discrete Contin. Dyn. Syst. 16, No. 4, 843–856 (2006; Zbl 1185.37043)]. The article under review presents the affirmative solution of the problem in the whole generality. Moreover, results presented here also confirm a conjecture stated in the article [Babenko and Bogatyj, loc.cit.] on bounded sequences satisfying Dold’s relations.

MSC:

37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
37C05 Dynamical systems involving smooth mappings and diffeomorphisms
55M20 Fixed points and coincidences in algebraic topology
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