Shadowing and Stability in p-adic dynamics

Abstract

In this paper, we study dynamical properties as shadowing and structural stability for a class of dynamics on $\mathbb{Z}_p$ and $\mathbb{Q}_p$, where $p \geq 2$ is a prime number. In particular, we prove that if $f: \mathbb{Z}_p \to \mathbb{Z}_p$ is a $(p^{-k},p^{m})$ ( $0 < m \leq k$ integers ) locally scaling map then $f$ is shadowing and structurally stable. We also study the number of conjugacy classes of these maps and we consider the above properties for $1$-Lipschitz maps of $\mathbb{Z}_p$ and for extensions of the shift map, contractions and dilatations on $\mathbb{Q}_p$.