Let
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{F}_q$
be the finite field of
$q$
elements. An analogue of the regular continued fraction expansion for an element
$\alpha $
in the field of formal Laurent series over
$\mathbb{F}_q$
is given uniquely by
$$\begin{equation*} \alpha = A_0(\alpha )+\cfrac {1}{A_1(\alpha )+\cfrac {1}{A_2(\alpha )+\ddots }}, \end{equation*}$$
where
$(A_n(\alpha ))_{n=0}^\infty $
is a sequence of polynomials with coefficients in
$\mathbb{F}_q$
such that
$\deg (A_n(\alpha ))\ge 1$
for all
$n\ge 1.$
We first prove the exactness of the continued fraction map in positive characteristic. This fact implies a number of strictly weaker properties. Particularly, we then use the weak-mixing property and ergodicity to establish various metrical results regarding the averages of partial quotients of continued fraction expansions. A sample result that we prove is that if
$(p_n)_{n=1}^\infty $
denotes the sequence of prime numbers, we have
$$\begin{equation*} \lim _{n\to \infty }\frac {1}{n}\sum _{j=1}^n \deg (A_{p_j}(\alpha )) = \frac {q}{q-1} \end{equation*}$$
for almost every
$\alpha $
with respect to Haar measure. In the case where the sequence
$(p_n)_{n=1}^\infty $
is replaced by
$(n)_{n=1}^\infty ,$
this result is due to V. Houndonougbo, V. Berthé and H. Nakada. Our proofs rely on pointwise subsequence and moving average ergodic theorems.