On Schneider\u2019s Continued Fraction Map on a Complete Non-Archimedean Field

Abstract

Let {mathcal {M}} denote the maximal ideal of the ring of integers of a non-Archimedean field K with residue class field k whose invertible elements, we denote k^{times }, and a uniformizer we denote pi . In this paper, we consider the map T_{v}: {mathcal {M}} rightarrow {mathcal {M}} defined by Tv(x)=πv(x)x-b(x),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} T_v(x) = \\frac{\\pi ^{v(x)}}{x} - b(x), \\end{aligned}$$\\end{document}where b(x) denotes the equivalence class to which frac{pi ^{v(x)}}{x} belongs in k^{times }. We show that T_v preserves Haar measure mu on the compact abelian topological group {mathcal {M}}. Let {mathcal {B}} denote the Haar sigma -algebra on {mathcal {M}}. We show the natural extension of the dynamical system ({mathcal {M}}, {mathcal {B}}, mu , T_v) is Bernoulli and has entropy frac{#( k)}{#( k^{times })}log (#( k)). The first of these two properties is used to study the average behaviour of the convergents arising from T_v. Here for a finite set A its cardinality has been denoted by # (A). In the case K = {mathbb {Q}}_p, i.e. the field of p-adic numbers, the map T_v reduces to the well-studied continued fraction map due to Schneider.