Almost isometric flat spaces and perturbation-equivalence (Q972495)

Let \(X\) and \(Y\) be two metric spaces. An almost isometry from \(X\) to \(Y\) is a sequence \((f_n)_{n\geq0}\) such that each \(f_n:X\to Y\) is a \(\lambda_n\)-bi-Lipschitz surjection and \(\lim_{n\to\infty}\lambda_n=1\). The author constructs an interesting simple example of closed subsets of \(\mathbb R\) which are almost isometric but not isometric. Using a non-principial ultrafilter \(\mathcal U\) on \(\mathbb N\) and the ultra-powers with respect to \(\mathcal U\) the author introduces the concepts of an almost isometry relative to \(\mathcal U\), a perturbation equivalence, a rigidity, an almost rigidity and so on. Some results are also obtained in these terms. The most interesting of them are closely connected with the above mentioned example.