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On conditions under which isometries have bounded orbits. (English) Zbl 0558.54021
The main result is the following: Let M be a metric space each bounded subset of which is totally bounded, and let \(f: M\to M\) be nonexpansive (d(f(x),f(y))\(\leq d(x,y),x,y\in M)\). If for some \(z_ 0\in M\) the sequence \(\{f^ n(z_ 0)\}\) contains a bounded subsequence, then for every \(z\in M\) the sequence \(\{f^ n(z)\}\) is bounded. This answers in the affirmative a question raised by the reviewer in [ibid. 22, 229-232 (1971; Zbl 0214.492)] where it is asked if the same is true for an isometry f of a G-space onto itself. All G-spaces are finitely compact, hence bounded subsets of such spaces are compact.
Reviewer: W.A.Kirk

MSC:
54E40 Special maps on metric spaces
Keywords:
G-spaces
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