On Schneider’s Continued Fraction Map on a Complete Non-Archimedean Field



Nair, Radhakrishnan and Haddley, Alena
(2021) On Schneider’s Continued Fraction Map on a Complete Non-Archimedean Field. Arnold Mathematical Journal.

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Abstract

Let 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×, and a uniformizer we denote π. In this paper, we consider the map Tv:M→M defined by Tv(x)=πv(x)x−b(x), where b(x) denotes the equivalence class to which πv(x)x belongs in k×. We show that Tv preserves Haar measure μ on the compact abelian topological group M. Let B denote the Haar σ-algebra on M. We show the natural extension of the dynamical system (M,B,μ,Tv) is Bernoulli and has entropy #(k)#(k×)log(#(k)). The first of these two properties is used to study the average behaviour of the convergents arising from Tv. Here for a finite set A its cardinality has been denoted by #(A). In the case K=Qp, i.e. the field of p-adic numbers, the map Tv reduces to the well-studied continued fraction map due to Schneider.

Item Type: Article
Divisions: Faculty of Science and Engineering > School of Physical Sciences
Depositing User: Symplectic Admin
Date Deposited: 08 Oct 2021 15:08
Last Modified: 05 Mar 2022 22:10
DOI: 10.1007/s40598-021-00190-y
Open Access URL: https://doi.org/10.1007/s40598-021-00190-y
URI: https://livrepository.liverpool.ac.uk/id/eprint/3139753