Skip to main content
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 9
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Xuan, Zuxing 2014. On conformal measures of parabolic meromorphic functions. Discrete and Continuous Dynamical Systems - Series B, Vol. 20, Issue. 1, p. 249.

    Kotus, Janina and Urbański, Mariusz 2006. Geometry and dynamics of some meromorphic functions. Mathematische Nachrichten, Vol. 279, Issue. 13-14, p. 1565.

    Kotus, Janina and Urbański, Mariusz 2004. Geometry and ergodic theory of non-recurrent elliptic functions. Journal d'Analyse Mathématique, Vol. 93, Issue. 1, p. 35.

    Mauldin, R.Daniel and Urbański, Mariusz 2002. Fractal Measures for Parabolic IFS. Advances in Mathematics, Vol. 168, Issue. 2, p. 225.

    Peres, Yuval Simon, Károly and Solomyak, Boris 2000. Self-similar sets of zero Hausdorff measure and positive packing measure. Israel Journal of Mathematics, Vol. 117, Issue. 1, p. 353.

    DENKER, M. and ROHDE, S. 1999. ON HAUSDORFF MEASURES AND SBR MEASURES FOR PARABOLIC RATIONAL MAPS. International Journal of Bifurcation and Chaos, Vol. 09, Issue. 09, p. 1763.

    Stratmann, Bernd 1995. Fractal dimensions for Jarník limit sets of geometrically finite Kleinian groups; the semi-classical approach. Arkiv för matematik, Vol. 33, Issue. 2, p. 385.

    Urbański, Mariusz 1994. Rational functions with no recurrent critical points. Ergodic Theory and Dynamical Systems, Vol. 14, Issue. 02,

    Denker, Manfred and Urbański, Mariusz 1992. Ergodic Theory and Related Topics III.


Geometric measures for parabolic rational maps

  • M. Denker (a1) and M. Urbański (a2)
  • DOI:
  • Published online: 01 September 2008

Let h denote the Hausdorff dimension of the Julia set J(T) of a parabolic rational map T. In this paper we prove that (after normalisation) the h-conformal measure on J(T) equals the h-dimensional Hausdorff measure Hh on J(T), if h ≥ 1, and equals the h-dimensional packing measure Πh on J(T), if h ≤ 1. Moreover, if h < 1, then Hh = 0 and, if h > 1, then Πh(J(T)) = ∞.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[2]P. Blanchard . Complex analytic dynamics of the Riemann sphere. Bull. Amer. Math. Soc. 11 (1984), 85141.

[3]H. Brolin . Invariant sets under iteration of rational functions. Ark. Mat. 6 (1965), 103144.

[5]M. Denker & M. Urbański . Absolutely continuous invariant measures for expansive rational maps with rationally indifferent periodic points. Forum Math. 3 (1991), 561579.

[6]M. Guzmán . Differentiation of integrals in ℝn. Springer Lecture Notes in Mathematics 481, Springer Verlag, 1975.

[9]D. Sullivan . Conformal dynamical systems. In: Geometric Dynamics. Springer Lecture Notes in Mathematics1007, Springer Verlag, 1983, pp 725752.

[10]D. Sullivan . Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Math. 153 (1984), 259277.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *