A family of p-adic isometries, fixed points, and the number three

Abstract

We study a continuous family of norm-preserving isometries of the p-adic unit disk, given as interpolations of a common arithmetic function, the q-analog of the identity, for principal p-adic units q. We show that the fixed point set is trivial except when p=3 and q is a generator of the principal 3-adic units; in that case the isometry given by each q has a unique nontrivial 3-adic fixed point, varying homeomorphically with q.