On p-adic Mobius maps

الأصل · EN

In this paper, we study three aspects of the p-adic Möbius maps. One is the group PSL(2,Oₚ), another is the geometrical characterization of the p-adic Möbius maps and its application, and the other is different norms of the p-adic Möbius maps. Firstly, we give a series of equations of the p-adic Möbius maps in PSL(2,Oₚ) between matrix, chordal, hyperbolic and unitary aspects. Furthermore, the properties of PSL(2,Oₚ) can be applied to study the geometrical characterization, the norms, the decomposition theorem of p-adic Möbius maps, and the convergence and divergence of p-adic continued fractions. Secondly, we classify the p-adic Möbius maps into four types and study the geometrical characterization of the p-adic Möbius maps from the aspects of fixed points in P¹Ber and the invariant axes which yields the decomposition theorem of p-adic Möbius maps. Furthermore, we prove that if a subgroup of PSL(2,Cₚ) containing elliptic elements only, then all elements fix the same point in HBer 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 MartinF.G1,F.G2, Beardon and Short AI to the non-archimedean settings. These inequalities of p-adic Möbius 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 Möbius maps which can be viewed as a special cases of the work in CJE and the discrete criteria of the subgroups of PSL(2,Cₚ).