On continued fraction expansions of quadratic irrationals in positive characteristic

Abstract

Let $R={\mathbb{F}}_q[Y]$ be the ring of polynomials over a finite field ${\mathbb{F}}_q$, let $\hat{K}={\mathbb{F}}_q((Y^{-1}))$ be the field of formal Laurent series over ${\mathbb{F}}_q$, let $f\in \hat{K}$ 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 $P^{n}f$ 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 $({\mathrm{PGL}}_2,\hat{K})$ and, with $A$ the diagonal subgroup of ${\mathrm{PGL}}_2(\hat{K})$, the escape of mass phenomena of [7] for $A$-invariant probability measures on the compact $A$-orbits along Hecke rays in the moduli space ${\mathrm{PGL}}_2(R)\backslash {\mathrm{PGL}}_2(\hat{K})$.