## Abstract

Let R D F_{q}[Y ] be the ring of polynomials over a finite field F_{q}, let K D F_{q}((Y -^{1}) be the field of formal Laurent series over F_{q}, let f 2 K be a quadratic irrational over F_{q}.Y / and let P 2 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/ ^{y} , 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/n PGL_{2}.K/ ^{y}

Original language | English |
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Pages (from-to) | 81-105 |

Number of pages | 25 |

Journal | Groups, Geometry, and Dynamics |

Volume | 14 |

Issue number | 1 |

DOIs | |

State | Published - 2020 |

## Keywords

- Artin map
- Bruhat-Tits tree
- Bruhat–Tits tree
- Continued fraction expansion
- Hecke tree
- Positive characteristic
- Quadratic irrational
- continued fraction expansion
- positive characteristic

## All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Geometry and Topology