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Multiplicities of fixed points of holomorphic maps in several complex variables. (English) Zbl 1019.37010
Summary: Let \(\Delta^v\) be the unit ball in \(\mathbb{C}^v\) with center 0 (the origin of \(\mathbb{C}^v)\) and let \(F:\Delta^v\to \mathbb{C}^v\) be a holomorphic map with \(F(0)=0\). This paper studies the fixed point multiplicities at the origin 0 of the iterates \(F^i=F\circ \cdots \circ F\) \((i\) times), \(i=1,2,\dots\). This problem is easy when \(v=1\), but it is very complicated when \(v>1\). We study this problem in general.

MSC:
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
32F99 Geometric convexity in several complex variables
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
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