Skip to main content
SpringerLink
Log in

Hausdorff dimension of limit sets

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

We exhibit a class of Schottky subgroups of \(\mathbf {PU}(1,n)\) (\(n \ge 2\)) which we call well-positioned and show that the Hausdorff dimension of the limit set \(\Lambda _\Gamma \) associated with such a subgroup \(\Gamma \), with respect to the spherical metric on the boundary of complex hyperbolic n-space, is equal to the growth exponent \(\delta _\Gamma \). For general \(\Gamma \) we establish (under rather mild hypotheses) a lower bound involving the dimension of the Patterson–Sullivan measure along boundaries of complex geodesics. Our main tool is a version of the celebrated Ledrappier–Young theorem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Balogh, Z.M., Hoefer-Isenegger, R., Tyson, J.T.: Lifts of Lipschitz maps and horizontal fractals in the Heisenberg group. Ergod. Theory Dyn. Syst. 26(3), 621–651 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Balogh, Z.M., Tyson, J.T., Warhurst, B.: Sub-Riemannian vs. Euclidean dimension comparison and fractal geometry on Carnot groups. Adv. Math. 220(2), 560–619 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Becker, H., Kechris, A.S.: The Descriptive Set Theory of Polish Group Actions. London Mathematical Society Lecture Note Series, vol. 232. Cambridge University Press, Cambridge (1996)

  4. Benoist, Y., Quint, J.-F.: Mesures stationnaires et fermés invariants des espaces homogènes. Ann. Math. (2) 174(2), 1111–1162 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bishop, C.J., Jones, P.W.: Hausdorff dimension and Kleinian groups. Acta Math. 179(1), 1–39 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, J., Pesin, Y.: Dimension of non-conformal repellers: a survey. Nonlinearity 23(4), R93–R114 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Corlette, K.: Hausdorff dimensions of limit sets. I. Invent. Math. 102(3), 521–541 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dufloux, L.: Hausdorff Dimension of Limit Sets. Theses, Université Paris 13, October 2015. https://hal.archives-ouvertes.fr/tel-01293924

  9. Dufloux, L.: Projections of Patterson–Sullivan measures and the dichotomy of Mohammadi-Oh. preprint (2016)

  10. Einsiedler, M., Lindenstrauss, E.: Diagonal actions on locally homogeneous spaces. In: Homogeneous Flows, Moduli Spaces and Arithmetic, volume 10 of Clay Math. Proc., pp. 155–241. Am. Math. Soc., Providence, RI (2010)

  11. Falconer, K.J.: The Hausdorff dimension of self-affine fractals. Math. Proc. Camb. Philos. Soc. 103(2), 339–350 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  12. Goldman, W.M.: Complex Hyperbolic Geometry. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (1999). Oxford Science Publications

  13. Hersonsky, S., Paulin, F.: Diophantine approximation for negatively curved manifolds. Math. Z. 241(1), 181–226 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hochman, M.: Dynamics on fractals and fractal distributions. ArXiv e-prints, August (2010)

  15. Käenmäki, A., Rajala, T., Suomala, V.: Local homogeneity and dimensions of measures. ArXiv e-prints, March (2010)

  16. Ledrappier, F., Young, L.-S.: The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension. Ann. Math. (2) 122(3), 540–574 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ledrappier, F., Xie, J.-S.: Vanishing transverse entropy in smooth ergodic theory. Ergod. Theory Dyn. Syst. 31(4), 1229–1235 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ledrappier, F.: Entropie et principe variationnel pour le flot géodésique en courbure négative pincée. In: Géométrie ergodique, volume 43 of Monogr. Enseign. Math., pp. 117–144. Enseignement Math., Geneva (2013)

  19. Ledrappier, F., Lindenstrauss, E.: On the projections of measures invariant under the geodesic flow. Int. Math. Res. Not. 9, 511–526 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Margulis, G.A., Tomanov, G.M.: Invariant measures for actions of unipotent groups over local fields on homogeneous spaces. Invent. Math. 116(1–3), 347–392 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces, volume 44 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1995). Fractals and rectifiability

  22. Mohammadi, A., Oh, H.: Ergodicity of unipotent flows and Kleinian groups. J. Am. Math. Soc. 28(2), 531–577 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Paulin, F.: On the critical exponent of a discrete group of hyperbolic isometries. Differ. Geom. Appl. 7(3), 231–236 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  24. Roblin, T.: Ergodicité et équidistribution en courbure négative. Mém. Soc. Math. Fr. (N.S.), (95):vi+96 (2003)

  25. Stratmann, B.O.: The exponent of convergence of Kleinian groups; on a theorem of Bishop and Jones. In: Fractal Geometry and Stochastics III, Volume 57 of Progr. Probab., pp. 93–107. Birkhäuser, Basel (2004)

  26. Wingren, P.: Concerning a real-valued continuous function on the interval with graph of Hausdorff dimension \(2\). Enseign. Math. (2) 41(1–2), 103–110 (1995)

    MathSciNet  MATH  Google Scholar 

  27. Winter, D.: Mixing of frame flow for rank one locally symmetric spaces and measure classification. ArXiv e-prints, March (2014)

Download references

Author information

Authors and Affiliations

  1. Oulun Yliopisto, Oulu, Finland

    Laurent Dufloux

Authors
  1. Laurent Dufloux
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Laurent Dufloux.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dufloux, L. Hausdorff dimension of limit sets. Geom Dedicata 191, 1–35 (2017). https://doi.org/10.1007/s10711-017-0240-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-017-0240-2

Keywords

Mathematics Subject Classification

Search

Navigation