Abstract
Closed geodesics associated to conjugacy classes of hyperbolic matrices in SL(2, ℤ) can be coded in two different ways. The geometric code, with respect to a given fundamental region, is obtained by a construction universal for all Fuchsian groups, while the arithmetic code, given by ‘—’ continued fractions, comes from the Gauss reduction theory and is specific for SL(2, ℤ). In this paper we give a complete description of all closed geodesics for which the two codes coincide.
Similar content being viewed by others
References
Adler, R. and Flatto, L.: Cross section maps for geodesic flows, I: The Modular surface in, A. Katok (ed.), in: Ergodictheory and Dynamical Systems II (1979/80), Prog. in Math., Birkhäuser, Basel, 1982, pp. 103–161.
Artin, E.: Ein Mechanisches System mit quasiergodischen Bahnen, Collected Papers, Addison-Wesley, New York, 1965, pp. 499–501.
Hardy, G. H. and Wright, E. M.: An introduction to the Theory of Numbers, Oxford University Press, 1979.
Hedlund, G. A.: A metrically transitive group defined by the modular group, Amer. J. Math. 57 (1935), 668–678.
Katok, S.: Reduction theory for Fuchsian groups, Math. Ann. 273 (1986), 461–470.
Katok, S.: Fuchsian Groups, The University of Chicago Press, Chicago, 1992.
Katok, A. and Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, New York, 1995.
Morse, M.: Symbolic dynamics, Institute for Advanced Study Notes, Princeton, 1966, (unpublished).
Series, C.: Symbolic dynamics for geodesic flows, Acta Math. 146 (1981), 103–128.
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.
Zagier, D.: Zetafunktionen und quadratische Körper: eine Einführung in die höhere Zahlentheorie, Hochschultext, Springer, Berlin, 1981.
Author information
Authors and Affiliations
Additional information
The work of the author was partially supported by NSF grant DMS-9404136.
Rights and permissions
About this article
Cite this article
Katok, S. Coding of closed geodesics after Gauss and morse. Geom Dedicata 63, 123–145 (1996). https://doi.org/10.1007/BF00148213
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00148213