The theta group and the continued fraction expansion with even partial quotients Cornelis Kraaikamp Artur Lopes Article Received: 01 March 1994 Revised: 04 May 1995 DOI :
10.1007/BF00181695

Cite this article as: Kraaikamp, C. & Lopes, A. Geom Dedicata (1996) 59: 293. doi:10.1007/BF00181695
Abstract F. Schweiger introduced the continued fraction with even partial quotients. We will show a relation between closed geodesics for the theta group (the subgroup of the modular group generated by z +2 and -1 / z ) and the continued fraction with even partial quotients. Using thermodynamic formalism, Tauberian results and the above-mentioned relation, we obtain the asymptotic growth number of closed trajectories for the theta group. Several results for the continued fraction expansion with even partial quotients are obtained; some of these are analogous to those already known for the usual continued fraction expansion related to the modular group, but our proofs are by necessity in general technically more difficult.

Mathematics Subject Classification (1991) 58F11

Key words Riemann surfaces Fuchsian groups continued fraction expansion thermodynamic formalism even partial quotients Supported by The Netherlands Organization for Scientific Research (NWO).

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Authors and Affiliations Cornelis Kraaikamp Artur Lopes 1. Department of Mathematics Technical University Delft Delft The Netherlands 2. Instituto de Matemática Universidade Federal do Rio Grande do Sul Porto Alegre, RS Brazil