Local isometries of compact metric spaces
Aleksander Całka · Proceedings of the American Mathematical Society · 1982
By local isometries we mean mappings which locally preserve distances. A few of the main results are: 1. For each local isometry f f of a compact metric space ( M , ρ ) (M,\rho ) into itself there exists a unique decomposition of M M into disjoint open sets, M = M 0 f ∪ ⋯ ∪ M n f M = M_0^f \cup \cdots \cup M_n^f , ( 0 ⩽ n > ∞ ) (0 \leqslant n > \infty ) such that (i) f ( M 0 f ) = M 0 f f(M_0^f) = M_0^f , and (ii) f ( M i f ) = M i − 1 f f(M_i^f) = M_{i - 1}^f and M i f ≠ ∅ M_i^f e \emptyset for each i , 1 ⩽ i ⩽ n i, 1 \leqslant i \leqslant n