Skip to main content
×
Home
    • Aa
    • Aa

Geometric measures for parabolic rational maps

  • M. Denker (a1) and M. Urbański (a2)
Abstract
Abstract

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)) = ∞.

Copyright
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? *
×
MathJax

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 2 *
Loading metrics...

Abstract views

Total abstract views: 17 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 28th May 2017. This data will be updated every 24 hours.