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Fixed point indices and periodic points of holomorphic mappings. (English) Zbl 1110.32007
Let \(f\) be a germ of holomorphic self-maps of \(\mathbb C^n\) fixing the origin \(O\). If \(O\) is an isolated fixed point for \(f\) then one can define the A. Dold index [Invent. Math. 74, No. 3, 419–435 (1983; Zbl 0583.55001)] \(P_M(f,O)\) for each integer \(M\) such that \(O\) is an isolated fixed point for \(f^{\circ M}\). If \(g\) is a holomorphic map close to \(f\) then it has \(P_M(f,O)\) (counting multiplicity) periodic points of period \(M\) near \(O\).
In this paper the author proves that \(P_M(f,O)>0\) if and only if \(df_O\) has a periodic point of period \(M\).
The necessity of the condition on \(df_O\) has already been proved for smooth maps by M. Shub and D. Sullivan [Topology 13, 189–191 (1974; Zbl 0291.58014)], but the condition is known to be not sufficient if the map is not holomorphic.
The strategy of the proof is as follows. The author explicitly computes the Dold index in some special cases (proving the sufficiency of the condition on \(df_O\) in such cases). Then show that the general case can be reduced to the previous one by means of small perturbations and changes of coordinates. He also reproves the necessity of the condition in the holomorphic setting.

32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
Full Text: DOI
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