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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Apostol, T.: Introduction to Analytic Number Theory, Springer-Verlag, 1976.
Bauer, M. and Lopes, A.: A billiard with very slow decay of correlation, Preprint, 1994.
Brietzke, E. and Lopes, A.: Trace class properties of the Ruelle Operator associated to an indifferent fixed point, Preprint, 1994.
Hardy and Wright, Introduction to the Theory of Numbers, Oxford Univ. Press, 1979.
Hofbauer, F.: Examples for the non-uniqueness of the equilibrium state, Trans. Amer. Math. Soc. 228 (1977), 223–241.
Kraaikamp, C.: The distribution of some sequences connected with the nearest integer continued fraction, Indag. Math. 49 (1987) 177–191.
Kraaikamp, C.: A new class of continued fraction expansions, Acta Arith. LVII (1991), 1–39.
Lalley, S. P.: Renewal theorems in symbolic dynamics, with applications to geodesic flows, non-Euclidean tessellations and their fractal limits, Acta Math. 163, (1990), 1–55.
Lopes, A.: The zeta function, non-differentiability of pressure and the critical exponent of transition, Adv. in Math. 101 (1993), 133–165.
Mayer, D. H.: On a ζ function related to the continued fraction transformation, Bull. Soc. Math. France 104 (1976), 195–203.
Mayer, D. H.: Continued fractions and related transformations, in T. Bedford et al. (eds), Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces 1991, pp. 175–220.
Parry, W. and Pollicott, M.: An analogue of the prime number theorem for closed orbits of Axiom A flows, Ann. Math. 118 (1983), 573–591.
Parry, W. and Pollicott, M.: Zeta functions and the periodic orbit structure of hyperbolic dynamics, Asterisque 187–188 (1990).
T. Pignataro: Hausdorff dimension, spectral theory and applications to the quantization of geodesic flows on surfaces of constant negative curvature, thesis, Princeton University, 1984.
Pollicott, M.: Meromorphic extensions for the generalized zeta functions, Invent. Math. 85 (1986), 147–164.
Pollicott, M.: A complex Ruelle-Perron-Frobenius theorem and two counterexamples, Ergodic Theory Dynamical Systems 4 (1984) 135–146.
Pollicott, M.: Distribution of closed geodesics on the modular surface and quadratic irrationals, Bull. Soc. Math. France 114 (1986), 431–446.
Pollicott, M.: Closed geodesics and zeta functions, in T. Bedford et al. (eds), Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces 1991, pp. 153–172.
Pollicott, M.: Some applications of thermodynamic formalism to manifolds with constant negative curvature, Adv. in Math. 85 (1991), 161–192.
Ruelle, D.: Thermodynamic Formalism, Addison-Wesley, 1978.
Series, C.: The modular surface and continued fractions, J. London Math. Soc. (2) 31 (1985), 69–80.
Series, C.: Geometrical Markov coding of geodesics on surfaces of constant negative curvature, Ergodic Theory Dynamical Systems 6 (1986), 601–625.
Schweiger, F.: Continued fractions with odd and even partial quotients, Arbeitsber. Math. Inst. Univ. Salzburg 4 (1982), 59–70.
Schweiger, F.: On the approximation by continued fractions with odd and even partial quotients, Arbeitsber. Math. Inst. Univ. Salzburg 1–2 (1984), 105–114.
Sullivan, D.: Discrete conformal groups and measurable dynamics. Bull. Amer. Math. Soc. 6 (1982), 57–73.
Urbański, M.: Hausdorff dimension of invariant subsets for endomorphisms of the circle with an indifferent fixed point, J. London Math. Soc. (2). 40 (1989), 158–170.
Walters, P.: An Introduction to Ergodic Theory, Springer-Verlag, 1982.
Author information
Authors and Affiliations
Additional information
Supported by The Netherlands Organization for Scientific Research (NWO).
Rights and permissions
About this article
Cite this article
Kraaikamp, C., Lopes, A. The theta group and the continued fraction expansion with even partial quotients. Geom Dedicata 59, 293–333 (1996). https://doi.org/10.1007/BF00181695
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00181695