JournalsggdVol. 14, No. 1pp. 81–105

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 R=Fq[Y]R=\mathbb F_q[Y] be the ring of polynomials over a finite field Fq\mathbb F_q, let \whatK=Fq((Y1))\what K=\mathbb F_q((Y^{-1})) be the field of formal Laurent series over Fq\mathbb F_q, let f\whatKf\in\what K be a quadratic irrational over Fq(Y)\mathbb F_q(Y) and let PRP\in R 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 PnfP^nf as n+n \to +\infty, proving, in sharp contrast with the case of quadratic irrationals in R\mathbb R over Q\mathbb Q 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 (PGL2,\whatK)(\mathrm {PGL}_2,\what K) and, with AA the diagonal subgroup of PGL2(\whatK)\mathrm {PGL}_2(\what K), the escape of mass phenomena of [7] for AA-invariant probability measures on the compact AA-orbits along Hecke rays in the moduli space PGL2(R)\bs\PGL2(\whatK)\mathrm {PGL}_2(R)\bs\PGL_2(\what K).

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