The pointwise convergence of p-adic Möbius maps

Abstract

The convergence of linear fractional transformations is an important topic in mathematics. We study the pointwise convergence of p-adic Mobius maps, and classify the possibilities of limits of pointwise convergent sequences of Mobius maps acting on the projective line ℙ1(ℂp), where ℂp is the completion of the algebraic closure of ℚp. We show that if the set of pointwise convergence of a sequence of p-adic Mobius maps contains at least three points, the sequence of p-adic Mobius maps either converges to a p-adic Mobius map on the projective line ℙ1(ℂp), or converges to a constant on the set of pointwise convergence with one unique exceptional point. This result generalizes the result of Piranian and Thron (1957) to the non-archimedean settings.