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Holomorphic retracts in $$B_H^{\infty}$$. (English) Zbl 1091.37010
M. Budzynska and T. Kuczumow [Contemporary Mathematics 364, 35–40 (2004; Zbl 1091.58005)] proved that if $$\mathcal {F}$$ is a countable family of holomorphic ($$k_{ B_H^{\infty}}$$-nonexpansive) commuting self-mappings of $$B_H^{\infty}$$ with a nonempty common fixed-point set, then this fixed-point set is a $$k_{ B_H^{\infty}}$$-nonexpansive retract of $$B_H^{\infty}$$.
In the present paper, the authors prove a general result of the same type: they show that the common fixed-point set of a commuting family of holomorphic mappings in $$B_H^{\infty}$$ is either empty or a holomorphic set.
##### MSC:
 37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics 46T25 Holomorphic maps in nonlinear functional analysis 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H10 Fixed-point theorems 32A10 Holomorphic functions of several complex variables 58B12 Questions of holomorphy and infinite-dimensional manifolds
Zbl 1091.58005
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