Iterates of holomorphic and \(k_ D\)-nonexpansive mappings in convex domains in \({\mathbb{C}}^ n\)
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.
scientific article; zbMATH DE number 4198450
scientific article