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

and Applied Analysis 3 Theorem 1.2 Rassias 18 . Let X be a real normed linear space and Y a real complete normed linear space. Assume that f : X → Y is an approximately additive mapping for which there exist constants θ ≥ 0 and p, q ∈ R such that r p q / 1 and f satisfies the inequality ∥ ∥f ( x y ) − f x − f(y)∥∥ ≤ θ‖x‖p∥∥y∥∥q 1.5 for all x, y ∈ X. Then there exists a unique additive mapping L...

We prove generalized Hyres-Ulam-Rassias stability of the cubic functional equation f(kx + y) + f(kx − y) = k[f(x + y) + f(x − y)] + 2(k − k)f(x) for all k ∈ N and the quartic functional equation f(kx + y) + f(kx − y) = k[f(x + y) + f(x − y)] + 2k(k − 1)f(x)− 2(k − 1)f(y) for all k ∈ N in non-Archimedean normed spaces.