In this paper, we study three aspects of the $p-$adic M\"obius maps. One is the group $\mathrm{ PSL}(2,\mathcal{ O}_{ p})$, another is the geometrical characterization of the $p-$adic M\"obius maps and its application, and the other is different norms of the $p-$adic M\"obius maps. Firstly, we give a series of equations of the $p-$adic M\"obius maps in $\mathrm{ PSL}(2,\mathcal{ O}_{ p})$ between matrix, chordal, hyperbolic and unitary aspects. Furthermore, the properties of $\mathrm{ PSL}(2,\mathcal{ O}_{ p})$ can be applied to study the geometrical characterization, the norms, the decomposition theorem of $p-$adic M\"obius maps, and the convergence and divergence of $p-$adic continued fractions. Secondly, we classify the $p-$adic M\"obius maps into four types and study the geometrical characterization of the $p-$adic M\"obius maps from the aspects of fixed points in $\mathbb{ P}^{ 1}_{ Ber}$ and the invariant axes which yields the decomposition theorem of $p-$adic M\"obius maps. Furthermore, we prove that if a subgroup of $\mathrm{ PSL}(2,\mathbb{ C}_{ p})$ containing elliptic elements only, then all elements fix the same point in $\mathbb{ H}_{ Ber}$ without using the famous theorem--Cartan fixed point theorem, and this means that this subgroup has potentially good reduction. In the last part, we extend the inequalities obtained by Gehring and Martin\cite{ F.G1,F.G2}, Beardon and Short \cite{ AI} to the non-archimedean settings. These inequalities of $p$-adic M\"obius maps are between the matrix, chordal, three-point and unitary norms. This part of work can be applied to study the convergence of the sequence of $p-$adic M\"obius maps which can be viewed as a special cases of the work in \cite{ CJE} and the discrete criteria of the subgroups of $\mathrm{ PSL}(2,\mathbb{ C}_{ p})$.
Categories: math.DS math.GR