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 $p\ge 2$ is a prime number. In particular, we prove that if $f:{\mathbb{Z}}_p\to {\mathbb{Z}}_p$ is a $(p^{-k},p^m)$ ( $0<m\le k$ 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 ${\mathbb{Q}}_p$.

20 pages