Journal d’Analyse Mathématique

, Volume 93, Issue 1, pp 35–102

Geometry and ergodic theory of non-recurrent elliptic functions


    • Faculty of Mathematics and Information SciencesWarsaw University of Technology
  • Mariusz Urbański
    • Department of MathematicsUniversity of North Texas

DOI: 10.1007/BF02789304

Cite this article as:
Kotus, J. & Urbański, M. J. Anal. Math. (2004) 93: 35. doi:10.1007/BF02789304


We explore the class of elliptic functions whose critical points all contained in the Julia set are non-recurrent and whose ω-limit sets form compact subsets of the complex plane. In particular, this class comprises hyperbolic, subhyperbolic and parabolic elliptic maps. Leth be the Hausdorff dimension of the Julia set of such an elliptic functionf. We construct an atomlessh-conformal measurem and show that theh-dimensional Hausdorff measure of the Julia set off vanishes unless the Julia set is equal to the entire complex plane ℂ. Theh-dimensional packing measure is positive and is finite if and only if there are no rationally indifferent periodic points. Furthermore, we prove the existence of a (unique up to a multiplicative constant) σ-finitef-invariant measure μ equivalent tom. The measure μ is shown to be ergodic and conservative, and we identify the set of points whose open neighborhoods all have infinite measure μ. In particular, we show that ∞ is not among them.

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© The Hebrew University Magnes Press 2004