On two affine-like dynamical systems in a local field

Abstract

Let K be a local field with valuation v and residue field k . We study two dynamical systems defined on K that can be considered as affine. The first one is the dynamical system ( K , φ ) where φ ( x ) = x p h + a ( h ∈ N , a ∈ K , v ( a ) ⩾ 0 ) and p is the characteristic of k . We prove that the minimal subsets of ( K , φ ) are cycles. For K of finite characteristic, the action of the Carlitz module on K gives a dynamical system that is similar to an affine system in characteristic 0.