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Iterates of holomorphic and \(k_ D\)-nonexpansive mappings in convex domains in \({\mathbb{C}}^ n\). (English) Zbl 0726.32016

If \(D\subseteq {\mathbb{C}}^ n\) is a bounded domain, then let \(k_ D\) be its Kobayashi distance and let f: \(D\to D\) be a \(k_ D\)-nonexpansive mapping. It is known that every holomorphic mapping f: \(D\to D\) is \(k_ D\)-nonexpansive. Denote by \({\mathcal H}(D)\) (respectively \({\mathcal N}(D))\) the set of holomorphic (respectively nonexpansive) mappings of D into D; for \(f\in {\mathcal N}(D)\) denote by \(\Gamma\) (f) (respectively \(\Gamma '(f))\) the closure in \(C(D,{\mathbb{C}}^ n)\) of the iterates of f (respectively the set of subsequential limits of the iterates of f) in the topology of the uniform convergence on compact subsets of D.
Theorem 1. For D a bounded domain in \({\mathbb{C}}^ n\) let f: \(D\to D\) be a holomorphic \((k_ D\)-nonexpansive) mapping. Then the following are equivalent:
(i) f has a fixed point;
(ii) \(\Gamma\) (f)\(\subseteq {\mathcal H}(D)\) (\(\Gamma\) (f)\(\subseteq {\mathcal N}(D));\)
(iii) \(\Gamma '(f)\) contains \(g\in {\mathcal H}(D)\) (g\(\in {\mathcal N}(D));\)
(iv) There exists \(x_ 0\in D\) and \(\{i_ m\}\) such that the sequence \(\{f^{i_ m}(x_ 0)\}\) lies strictly inside D.
The structure of Fix(f) for f a \(k_ D\)-nonexpansive mapping, respectively the existence of the common fixed points for a family of \(k_ D\)-nonexpansive mappings are also considered.

MSC:

32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
32A10 Holomorphic functions of several complex variables
32A17 Special families of functions of several complex variables
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