Topological conjugacy between Schneider's continued fraction map and a shift map on ${\mathbb{Q}}_p$

We prove that Schneider’s continued fraction map is topologically conjugate to a shift map defined on ${\mathbb{Q}}_p$, and the topological conjugation $f\text{\hspace{0.17em}⁣}:{\mathbb{Q}}_p\to {\mathbb{Q}}_p$ is an isometry such that $f(\mathbb{Q})=\mathbb{Z}[\frac{1}{p}]$. Furthermore, for $x\in {\mathbb{Q}}_p$ we proved that $x\in \mathbb{Q}$ if and only if $f(x)\in \mathbb{Z}[\frac{1}{p}]$.