# 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

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## Abstract

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

## 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