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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
##### Keywords:
holomorphic map; fixed point multiplicities
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##### References:
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