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Geometry and ergodic theory of non-recurrent elliptic functions

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Abstract

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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Authors and Affiliations

  1. Faculty of Mathematics and Information Sciences, Warsaw University of Technology, 00-661, Warsaw, Poland

    Janina Kotus

  2. Department of Mathematics, University of North Texas, P.O. Box 311430, 76203-1430, Denton, TX, USA

    Mariusz Urbański

Authors
  1. Janina Kotus
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  2. Mariusz Urbański
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Corresponding author

Correspondence to Janina Kotus.

Additional information

The research of the first author was supported in part by the Foundation for Polish Science, the Polish KBN Grant No 2 PO3A 034 25 and TUW Grant no 503G 112000442200. She also wishes to thank the University of North Texas where this research was conducted.

The research of the second author was supported in part by the NSF Grant DMS 0100078. Both authors were supported in part by the NSF/PAN grant INT-0306004.

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Kotus, J., Urbański, M. Geometry and ergodic theory of non-recurrent elliptic functions. J. Anal. Math. 93, 35–102 (2004). https://doi.org/10.1007/BF02789304

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  • DOI: https://doi.org/10.1007/BF02789304

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