On continued fraction expansions of quadratic irrationals in positive characteristic

  • Frédéric Paulin

    Université Paris-Sud, Orsay, France
  • Uri Shapira

    Technion - Israel Institute of Technology, Haifa, Israel
On continued fraction expansions of quadratic irrationals in positive characteristic cover

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Abstract

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 [1], that they have one such degree very large with respect to the other ones. We use arguments of [2] 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 [7] for -invariant probability measures on the compact -orbits along Hecke rays in the moduli space .

Cite this article

Frédéric Paulin, Uri Shapira, On continued fraction expansions of quadratic irrationals in positive characteristic. Groups Geom. Dyn. 14 (2020), no. 1, pp. 81–105

DOI 10.4171/GGD/535