Abstract
Let R D Fq[Y ] be the ring of polynomials over a finite field Fq, let K D Fq((Y -1) be the field of formal Laurent series over Fq, let f 2 K be a quadratic irrational over Fq.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 .PGL2; K/ and, with A the diagonal subgroup of PGL2.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 PGL2.R/n PGL2.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