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

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

$$\begin{aligned} T_v(x) = \frac{\pi ^{v(x)}}{x} - b(x), \end{aligned}$$

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.