# 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

## Abstract

Let $R=F_{q}[Y]$ be the ring of polynomials over a finite field $F_{q}$, let $K=F_{q}((Y_{−1}))$ be the field of formal Laurent series over $F_{q}$, let $f∈K$ be a quadratic irrational over $F_{q}(Y)$ and let $P∈R$ 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 $n→+∞$, proving, in sharp contrast with the case of quadratic irrationals in $R$ over $Q$ 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 $(PGL_{2},K)$ and, with $A$ the diagonal subgroup of $PGL_{2}(K)$, the escape of mass phenomena of [7] for $A$-invariant probability measures on the compact $A$-orbits along Hecke rays in the moduli space $PGL_{2}(R)\PGL_{2}(K)$.

## 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