Let F_q be the finite field of q elements. An analogue of the regular continued fraction expansion for an element in the field of formal Laurent series over F_q is given uniquely by where (An.(α))_(n=0)~∞ is a sequence of polynomials with coefficients in F_q such that deg.A_n(α))≥ 1 for all n > 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 (pn)_(n=1)~∞ denotes the sequence of prime numbers, we have for almost every with respect to Haar measure. In the case where the sequence (pn)_(n=1)~∞ is replaced by (n)_(n=1)~∞ this result is due to V. Houndonougbo, V. Berthé and H. Nakada. Our proofs rely on pointwise subsequence and moving average ergodic theorems.
展开▼
