  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 $\mathbb{Q}_p$.
