Isometries of ultrametric normed spaces

TL;DR

This paper explores the structure of isometries in ultrametric normed spaces, revealing their fractal nature and contrasting their properties with classical real normed spaces, especially regarding linearity of isometries.

Contribution

It introduces a fractal perspective on isometries of ultrametric spaces and investigates their implications for classical problems like Mazur's rotation problem and Tingley's problem.

Findings

Isometries form a fractal-like group in ultrametric spaces.

Ultrametric isometries can be highly non-linear.

Contrasts with linear isometries in real normed spaces.

Abstract

We show that the group of isometries of an ultrametric normed space can be seen as a kind of a fractal. Then, we apply this description to study ultrametric counterparts of some classical problems in Archimedean analysis, such as the so called Probl\`eme des rotations de Mazur or Tingley's problem. In particular, it turns out that, in contrast with the case of real normed spaces, isometries between ultrametric normed spaces can be very far from being linear.