On local isometries of finitely compact metric spaces

Aleksander Całka

Vol. 103 (1982), No. 2, 337–345

DOI: 10.2140/pjm.1982.103.337

Abstract

By local isometries we mean mappings which locally preserve distances. Local isometries which do not increase distances are called nonexpansive local isometries. A few of the main results are:

1. Let f be a local isometry (nonexpansive local isometry) of a finitely compact metric space (M,ρ) into itself. If for each (some) z ∈ M the sequence {fⁿ(z)} is bounded, then there exists a unique decomposition of M into disjoint open sets, M = M₀^f ∪ M₁^f ∪⋯ , such that (i) f maps M₀^f injectively into itself, and (ii) f(M_i+1^f) ⊂ M_i^f for each i = 0,1,⋯ . Moreover, f maps M₀^f homeomorphically (isometrically) onto itself.
2. Let f be a nonexpansive local isometry (local isometry) of a connected (convex) finitely compact metric space (M,ρ) into itself. If for some z ∈ M the sequence {fⁿ(z)} is bounded, then f is an isometry onto.

Mathematical Subject Classification 2000

Primary: 54E40

Secondary: 54E45

Milestones

Received: 5 January 1981

Revised: 20 September 1981

Published: 1 December 1982