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Journal of the London Mathematical Society Advance Access originally published online on October 15, 2007
Journal of the London Mathematical Society 2007 76(2):399-418; doi:10.1112/jlms/jdm063
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© 2007 London Mathematical Society

Dynamique sur le rayon modulaire et fractions continues en caractéristique p

Anne Broise-Alamichel

Université Paris-Sud 11
Laboratoire de Mathématique
UMR 8628 du CNRS
Bâtiment 425
Équipe de Topologie et Dynamique
91405 Orsay cedex
France
anne.broise{at}math.u-psud.fr

Frédéric Paulin

Département de Mathématique et Applications
UMR 8553 du CNRS
Ecole Normale Supérieure
45 rue d’Ulm
75230 Paris cedex 05
France

Let K be the field of formal Laurent series in X–1 over the finite field k, and let A be the ring of polynomials in X over k. One of the main results of the paper is to give a natural coding of the (discrete) geodesic flow on the quotient of the Bruhat–Tits treeT of PGL2(K) by PGL2(A), using the continued fraction expansion of the endpoints of the geodesic lines in T (the space of ends of T identifies with P 1(K). In particular, the invariance of the Haar measure by the Artin transformation can be deduced from the invariance of the Bowen–Margulis measure by the geodesic flow.

2000 Mathematics Subject Classification 11J70, 20G25, 20E08, 37A45, 11K50.

Received September 18, 2005; revised October 26, 2006; published online October 15, 2007.


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