Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-30T20:47:52.351Z Has data issue: false hasContentIssue false
  • English
  • Français

Transitivity for the modular group

Published online by Cambridge University Press:  24 October 2008

Mark Sheingorn
Mark Sheingorn
Affiliation:
Baruch College, CUNY, New York, NY 10010
Get access Rights & Permissions [Opens in a new window]

Extract

Let Γ be a Fuchsian group of the first kind acting on the upper half plane H+. Let be a Ford fundamental region for Γ in H+. Let ξ be a real number (a limit point) and let L( = Lξ) = {ξ + iy|0 ≤ y < 1}. L can be broken into successive intervals each one of which can be mapped by an element of Γ into . Since L is a hyperbolic line (h-line), this gives us a set of h-arcs in which we will call the image.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 88 , Issue 3 , November 1980 , pp. 409 - 423
Copyright
Copyright © Cambridge Philosophical Society 1980

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Artin, E.Ein mechanisches System mit quasiergodischen Bahnen, Abh. Math. Sem. Univ. Hamburg 3 (1923), 170–175.CrossRefGoogle Scholar
(2)Beardon, A. and Maskit, B.Limit points of Kleinian groups and finite sided fundamental polyhedra, Acta Math. 132 (1974), 1–12.CrossRefGoogle Scholar
(3)Cusick, T. W.Sums and products of continued fractions, Proc. Amer. Math. Soc. 27 (1971), 35–38.CrossRefGoogle Scholar
(4)Divis, B.On the sums of continued fractions, Acta Arith. 22 (1973), 157–173.CrossRefGoogle Scholar
(5)Hall, M. JrOn the sum and product of continued fractions, Ann. of Math. (2) 48 (1947), 966–993.CrossRefGoogle Scholar
(6)Hedlund, G. A.The dynamics of geodesic flows, Bull. Amer. Math. Soc. 45 (1939), 241–260.CrossRefGoogle Scholar
(7)Hopf, E.Fuchsian groups and ergodic theory. Trans. Amer. Math. Soc. 39 (1939), 299–314.CrossRefGoogle Scholar
(8)Kaufman, R.On Hausdorff dimension of projections, Mathematika 15 (1968), 153–155.CrossRefGoogle Scholar
(9)Kaufman, R.An exceptional set for Hausdorff dimension, Mathematika 16 (1969), 57–58.CrossRefGoogle Scholar
(10)Khinchin, A. Ya.Continued Fractions (University of Chicago Press, 1964).Google Scholar
(11)Lehner, J.Discontinuous Groups and Automorphic Functions, Math. Survey no. 8, Amer. Math. Soc., 1964, Providence.CrossRefGoogle Scholar
(12)Masur, H. Dense geodesics in Teichmüller space. (To appear.)Google Scholar
(13)Myrberg, P. J.Einige Andwendungen der Kettenbrüche in der Theorie der binären quadratischen Formen und der elliptischen Modulfunktionen, Ann. Acad. Sci. Fenn Ser. AI 23 (1924).Google Scholar
(14)Myrberg, P. J.Ein Approximationssatz für die Fuchsschen Gruppen, Acta Math. 57 (1931), 389–409.CrossRefGoogle Scholar
(15)Nicholls, P.Special limit points for Fuchsian groups and automorphic functions near the limit set, Indiana Univ. Math. J. 24 (1974), 143–148.CrossRefGoogle Scholar
(16)Nicholls, P.Transitivity properties for Fuchsian groups, Can. J. Math. 28 (1976), 805–814.CrossRefGoogle Scholar
(17)Rankin, R. A.Modular Forms and Functions (Cambridge University Press, 1977).CrossRefGoogle Scholar
(18)Rosen, D.A class of continued fractions associated with certain properly discontinuous groups, Duke Math. J. 21 (1954), 549–563.CrossRefGoogle Scholar
(19)Sheingorn, M.Boundary behavior of automorphic forms and transitivity for the modular group, Ill. J. Math. (To appear.)Google Scholar