A Note on the Algebraic Growth Rate of Poincaré Series for Kleinian Groups

Conference paper
Part of the Springer Proceedings in Mathematics book series (PROM, volume 9)

Abstract

In this note, we employ infinite ergodic theory to derive estimates for the algebraic growth rate of the Poincaré series for a Kleinian group at its critical exponent of convergence.

Dedicated to S.J. Patterson on the occasion of his 60th birthday.

Poincaré series Infinte ergodic theory Kleinian groups 

References

  1. 1.
    J. Aaronson. An introduction to infinite ergodic theory. Mathematical Surveys and Monographs 50, American Mathematical Society, 1997.Google Scholar
  2. 2.
    J. Aaronson, M. Denker, M. Urbański. Ergodic theory for Markov fibred systems and parabolic rational maps. Trans. Amer. Math. Soc. 337 (2): 495–548, 1993.Google Scholar
  3. 3.
    A. F. Beardon. The exponent of convergence of Poincaré series. Proc. London Math. Soc. (3) 18: 461–483, 1968.Google Scholar
  4. 4.
    R. Bowen, C. Series. Markov maps associated with Fuchsian groups. Publ. Math., Inst. Hautes Etud. Sci. 50:153–170, 1979.Google Scholar
  5. 5.
    M. Denker, B.O. Stratmann. Patterson measure: classics, variations and applications.Google Scholar
  6. 6.
    D. Epstein, C. Petronio. An exposition of Poincare’s polyhedron theorem. Enseign. Math., II. Ser. 40: 113–170, 1994.Google Scholar
  7. 7.
    J. Fiala, P. Kleban. Intervals between Farey fractions in the limit of infinite level. Ann. Sci. Math. Qubec 34 (1), 63–71, 2010.Google Scholar
  8. 8.
    M. Kesseböhmer, B.O. Stratmann. A multifractal formalism for growth rates and applications to geometrically finite Kleinian groups. Ergod. Th. & Dynam. Sys., 24 (01):141–170, 2004.MATHCrossRefGoogle Scholar
  9. 9.
    M. Kesseböhmer, B.O. Stratmann. On the Lebesgue measure of sum-level sets for continued fractions. To appear in Discrete Contin. Dyn. Syst. Google Scholar
  10. 10.
    M. Kesseböhmer, B.O. Stratmann. A dichotomy between uniform distributions of the Stern-Brocot and the Farey sequence. To appear in Unif. Distrib. Theory. 7 (2), 2011.Google Scholar
  11. 11.
    P. Nicholls. The ergodic theory of discrete groups, London Math. Soc. Lecture Note Series 143, Cambr. Univ. Press, Cambridge, 1989.Google Scholar
  12. 12.
    S.J. Patterson. The limit set of a Fuchsian group. Acta Math. 136 (3–4): 241–273, 1976.Google Scholar
  13. 13.
    C. Series. The modular surface and continued fractions. J. London Math. Soc. (2) 31 (1): 69–80, 1985.Google Scholar
  14. 14.
    C. Series. Geometrical Markov coding of geodesics on surfaces of constant negative curvature. Ergod. Th. & Dynam. Sys. 6: 601–625, 1986.Google Scholar
  15. 15.
    M. Stadlbauer. The Bowen–Series map for some free groups. Dissertation 2002, University of Göttingen. Mathematica Gottingensis 5: 1–53, 2002.Google Scholar
  16. 16.
    M. Stadlbauer. The return sequence of the Bowen–Series map associated to punctured surfaces. Fundamenta Math. 182: 221–240, 2004.Google Scholar
  17. 17.
    M. Stadlbauer, B.O. Stratmann. Infinite ergodic theory for Kleinian groups. Ergod. Th. & Dynam. Sys. 25: 1305–1323, 2005.Google Scholar
  18. 18.
    B.O. Stratmann, S.L. Velani. The Patterson measure for geometrically finite groups with parabolic elements, new and old. Proc. London Math. Soc. (3) 71 (1): 197–220, 1995.Google Scholar
  19. 19.
    D. Sullivan. The density at infinity of a discrete group of hyperbolic motions. Inst. Haut. Études Sci. Publ. Math. 50: 171–202, 1979.Google Scholar
  20. 20.
    D. Sullivan. Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Math. 153: 259–277, 1984.Google Scholar
  21. 21.
    D. Sullivan. Discrete conformal groups and measurable dynamics. Bull. Amer. Math. Soc. 6: 57–73, 1982.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Fachbereich 3 – Mathematik und InformatikUniversität BremenBremenGermany

Personalised recommendations