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Let be the ring of polynomials over a finite field , let be the field of formal Laurent series over , let be a quadratic irrational over and let be an irreducible polynomial. We study the asymptotic properties of the degrees of the coefficients of the continued fraction expansion of quadratic irrationals such as as , proving, in sharp contrast with the case of quadratic irrationals in over considered in , that they have one such degree very large with respect to the other ones. We use arguments of  giving a relationship with the discrete geodesic flow on the Bruhat–Tits building of and, with the diagonal subgroup of , the escape of mass phenomena of  for -invariant probability measures on the compact -orbits along Hecke rays in the moduli space .
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Frédéric Paulin, Uri Shapira, On continued fraction expansions of quadratic irrationals in positive characteristic. Groups Geom. Dyn. 14 (2020), no. 1, pp. 81–105