On two affine-like dynamical systems in a local field

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)=xph+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.