Duke Mathematical Journal

On the uniform equidistribution of long closed horocycles

Andreas Strömbergsson

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

It is well known that on any given hyperbolic surface of finite area, a closed horocycle of length becomes asymptotically equidistributed as →∞. In this paper we prove that any subsegment of length greater than 1/2 + ε of such a closed horocycle also becomes equidistributed as →∞. The exponent 1/2 + ε is the best possible and improves upon a recent result by Hejhal [He3]. We give two proofs of the above result; our second proof leads to explicit information on the rate of convergence. We also prove a result on the asymptotic joint equidistribution of a finite number of distinct subsegments having equal length proportional to .

Article information

Source
Duke Math. J., Volume 123, Number 3 (2004), 507-547.

Dates
First available in Project Euclid: 11 June 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1086957715

Digital Object Identifier
doi:10.1215/S0012-7094-04-12334-6

Mathematical Reviews number (MathSciNet)
MR2068968

Zentralblatt MATH identifier
1060.37023

Subjects
Primary: 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
Secondary: 11F 30F35: Fuchsian groups and automorphic functions [See also 11Fxx, 20H10, 22E40, 32Gxx, 32Nxx]

Citation

Strömbergsson, Andreas. On the uniform equidistribution of long closed horocycles. Duke Math. J. 123 (2004), no. 3, 507--547. doi:10.1215/S0012-7094-04-12334-6. https://projecteuclid.org/euclid.dmj/1086957715


Export citation

References

  • A. Alvarez-Parrilla, Asymptotic relations among Fourier coefficients of real-analytic Eisenstein series, Trans. Amer. Math. Soc. 352 (2000), 5563–5582.
  • C. B. Balogh, Uniform asymptotic expansions of the modified Bessel function of the third kind of large imaginary order, Bull. Amer. Math. Soc. 72 (1966), 40–43.
  • –. –. –. –., Asymptotic expansions of the modified Bessel function of the third kind of imaginary order, SIAM J. Appl. Math. 15 (1967), 1315–1323.
  • I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinaĭ, Ergodic Theory, Grundlehren Math. Wiss. 245, Springer, New York, 1982.
  • S. G. Dani, Invariant measures of horospherical flows on noncompact homogeneous spaces, Invent. Math. 47 (1978), 101–138.
  • –. –. –. –., Invariant measures and minimal sets of horospherical flows, Invent. Math. 64 (1981), 357–385.
  • S. G. Dani and G. A. Margulis, “Limit distributions of orbits of unipotent flows and values of quadratic forms” in I. M. Gel'fand Seminar, Adv. Soviet Math. 16, Part I, Amer. Math. Soc., Providence, 1993, 91–137.
  • A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Vol. II, McGraw-Hill, New York, 1953.
  • A. Eskin and C. McMullen, Mixing, counting, and equidistribution in Lie groups, Duke Math. J. 71 (1993), 181–209.
  • W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II, Wiley, New York, 1966.
  • L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J. 119 (2003), 465–526.
  • J. L. Hafner, Some remarks on odd Maass wave forms (and a correction to “Zeros of $L$-functions attached to Maass forms” [by C. Epstein, J. L. Hafner, and P. Sarnak, Math. Z. 190 (1985), 113–128.; ]), Math. Z. 196 (1987), 129–132.
  • D. A. Hejhal, The Selberg Trace Formula for $\PSL(2,\mathbbR)$, Vol. 2, Lecture Notes in Math. 1001, Springer, Berlin, 1983.
  • –. –. –. –., “On value distribution properties of automorphic functions along closed horocycles” in XVIth Rolf Nevanlinna Colloquium (Joensuu, Finland, 1995), de Gruyter, Berlin, 1996, 39–52.
  • –. –. –. –., “On the uniform equidistribution of long closed horocycles” in Loo-Keng Hua: A Great Mathematician of the Twentieth Century, Asian J. Math. 4, Int. Press, Somerville, Mass., 2000, 839–853.
  • H. Iwaniec, Introduction to the Spectral Theory of Automorphic Forms, Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoamericana, Madrid, 1995.
  • W. B. Jurkat and J. W. Van Horne, The uniform central limit theorem for theta sums, Duke Math. J. 50 (1983), 649–666.
  • A. Leutbecher, Über die Heckeschen Gruppen $\mathbbG(\lambda)$, Abh. Math. Sem. Univ. Hamburg 31 (1967), 199–205.
  • A. Manning, “Dynamics of geodesic and horocycle flows on surfaces of constant negative curvature” in Ergodic Theory, Symbolic Dynamics, and Hyperbolic Spaces (Trieste, Italy, 1989), Oxford Sci. Publ., Oxford Univ. Press, New York, 1991, 71–91.
  • G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Ergeb. Math. Grenzgeb (3) 17, Springer, Berlin, 1991.
  • J. Marklof, Limit theorems for theta sums, Duke Math. J. 97 (1999), 127–153.
  • –. –. –. –., “Theta sums, Eisenstein series, and the semiclassical dynamics of a precessing spin” in Emerging Applications of Number Theory (Minneapolis, 1996), IMA Vol. Math. Appl. 109, Springer, New York, 1999, 405–450.
  • T. Miyake, Modular Forms, Springer, Berlin, 1989.
  • S. J. Patterson, Diophantine approximation in Fuchsian groups, Philos. Trans. Roy. Soc. London Ser. A 282 (1976), 527–563.
  • M. Ratner, On Raghunathan's measure conjecture, Ann. of Math. (2) 134 (1991), 545–607.
  • –. –. –. –., Raghunathan's topological conjecture and distributions of unipotent flows, Duke Math. J. 63 (1991), 235–280.
  • –. –. –. –., Raghunathan's conjectures for $\SL(2,\mathbbR)$, Israel J. Math. 80 (1992), 1–31.
  • P. Sarnak, Asymptotic behavior of periodic orbits of the horocycle flow and Eisenstein series, Comm. Pure Appl. Math. 34 (1981), 719–739.
  • N. Shah, Limit distributions of expanding translates of certain orbits on homogeneous spaces, Proc. Indian Acad. Sci. Math. Sci. 106 (1996), 105–125.
  • H. Shimizu, On discontinuous groups operating on the product of the upper half planes, Ann. of Math. (2) 77 (1963), 33–71.
  • G. Shimura, Introduction to the Arithmetic Theory of Automorphic Forms, Kanô Memorial Lectures 1, Publ. Math. Soc. Japan 11, Iwanami Shoten, Tokyo; Princeton Univ. Press, 1971.
  • A. Strömbergsson, “Some results on the uniform equidistribution of long closed horocycles” in Studies in the Analytic and Spectral Theory of Automorphic Forms, Ph.D. thesis, Uppsala University, Uppsala, Sweden, 2001, 137–226. http://www.math.uu.se/$\sim$astrombe/papers.html
  • G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge Univ. Press, Cambridge, 1944.
  • S. A. Wolpert, Semiclassical limits for the hyperbolic plane, Duke Math. J. 108 (2001), 449–509.
  • –. –. –. –., Asymptotic relations among Fourier coefficients of automorphic eigenfunctions, Trans. Amer. Math. Soc. 356 (2004), 427–456.
  • D. Zagier, “Eisenstein series and the Riemann zeta function” in Automorphic Forms, Representation Theory and Arithmetic (Bombay, 1979), Tata Inst. Fund. Res. Studies in Math. 10, Tata Inst. Fundamental Res., Bombay, 1981, 275–301.