Minimal subsystems of affine dynamics on local fields

Abstract

We describe the dynamics of an arbitrary affine dynamical system on a local field by exhibiting all its minimal subsystems. In the special case of the field \({\mathbb{Q}_p}\) of p-adic numbers, for any non-trivial affine dynamical system, we prove that the field \({\mathbb{Q}_p}\) is decomposed into a countable number of invariant balls or spheres each of which consists of a finite number of minimal subsets. Consequently, we give a complete classification of topological conjugacy for non-trivial affine dynamics on \({\mathbb{Q}_p}\) . For each given prime p, there is a finite number of conjugacy classes.