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    Mayer, Dieter Mühlenbruch, Tobias and Strömberg, Fredrik 2012. The transfer operator for the Hecke triangle groups. Discrete and Continuous Dynamical Systems, Vol. 32, Issue. 7, p. 2453.


    Schmidt, Thomas A. and Sheingorn, Mark 1995. Length spectra of the Hecke triangle groups. Mathematische Zeitschrift, Vol. 220, Issue. 1, p. 369.


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  • Bulletin of the Australian Mathematical Society, Volume 46, Issue 3
  • December 1992, pp. 459-474

Hecke groups and continued fractions

  • David Rosen (a1) and Thomas A. Schmidt (a2)
  • DOI: http://dx.doi.org/10.1017/S0004972700012120
  • Published online: 01 April 2009
Abstract

The Hecke groups

are Fuchsian groups of the first kind. In an interesting analogy to the use of ordinary continued fractions to study the geodesics of the modular surface, the λ-continued fractions (λF) introduced by the first author can be used to study those on the surfaces determined by the Gq. In this paper we focus on periodic continued fractions, corresponding to closed geodesics, and prove that the period of the λF for periodic has nearly the form of the classical case. From this, we give: (1) a necessary and sufficient condition for to be periodic; (2) examples of elements of ℚ(λq) which also have such periodic expansions; (3) a discussion of solutions to Pell's equation in quadratic extensions of the ℚ(λq); and (4) Legendre's constant of diophantine approximation for the Gq, that is, γq such that < γq/Q2 implies that P/Q of “reduced finite λF form” is a convergent of real α ∉ Gq(∞).

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[2]R. Adler and L. Flatto , ‘Geodesic flows, interval maps, and symbolic dynamics’, Bull. AMS 25 (1991), 229334.

[3]E. Artin , ‘Ein mechanisches System mit quasiergodischen Bahnen’, Abh. Math. Sem. Univ. Hamburg 3 (1924), 170175.

[4]A. F. Beardon , The Geometry of discrete groups (Springer-Verlag, Berlin, Heidelberg, New York, 1983).

[8]M.C. Gutzwiller , Chaos in classical and quantum mechanics (Springer-Verlag, Berlin, Heidelberg, New York, 1990).

[10]G. Hedlund , ‘On the metrical transitivity of the geodesics on closed surfaces of constant negative curvature’, Ann. of Math (2)35 (1934), 787808.

[13]J. Lehner , ‘The local Hurwitz constant and Diophantine approximation on Hecke groups’, Math Comp 55 (1990), 765781.

[14]A. Leutbecher , ‘Über die Heckeschen Gruppen G(λ) II’, Math Ann. 211 (1974), 6384.

[15]D. Rosen , ‘A class of continued fractions associated to certain properly discontinuous groups’, Duke Math. J. 21 (1954), 549562.

[17]D. Rosen , ‘The substitutions of the Hecke group г(2cos π/5)’, Arch. Math. 46 (1986), 533538.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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