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 $\what K=\mathbb F\_q((Y^{-1}))$ 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 $\mathrm {PGL}\_2(R)\bs\PGL\_2(\what K)$.