AbstractLet $${\mathcal {M}}$$
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 $$k^{\times }$$
k
×
, and a uniformizer we denote $$\pi $$
π
. In this paper, we consider the map $$T_{v}: {\mathcal {M}} \rightarrow {\mathcal {M}}$$
T
v
:
M
→
M
defined by $$\begin{aligned} T_v(x) = \frac{\pi ^{v(x)}}{x} - b(x), \end{aligned}$$
T
v
(
x
)
=
π
v
(
x
)
x
-
b
(
x
)
,
where b(x) denotes the equivalence class to which $$\frac{\pi ^{v(x)}}{x}$$
π
v
(
x
)
x
belongs in $$k^{\times }$$
k
×
. We show that $$T_v$$
T
v
preserves Haar measure $$\mu $$
μ
on the compact abelian topological group $${\mathcal {M}}$$
M
. Let $${\mathcal {B}}$$
B
denote the Haar $$\sigma $$
σ
-algebra on $${\mathcal {M}}$$
M
. We show the natural extension of the dynamical system $$({\mathcal {M}}, {\mathcal {B}}, \mu , T_v)$$
(
M
,
B
,
μ
,
T
v
)
is Bernoulli and has entropy $$\frac{\#( k)}{\#( k^{\times })}\log (\#( k))$$
#
(
k
)
#
(
k
×
)
log
(
#
(
k
)
)
. The first of these two properties is used to study the average behaviour of the convergents arising from $$T_v$$
T
v
. Here for a finite set A its cardinality has been denoted by $$\# (A)$$
#
(
A
)
. In the case $$K = {\mathbb {Q}}_p$$
K
=
Q
p
, i.e. the field of p-adic numbers, the map $$T_v$$
T
v
reduces to the well-studied continued fraction map due to Schneider.
On Schneider’s Continued Fraction Map on a Complete Non-Archimedean Field