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Hausdorff measures on Julia sets of subexpanding rational maps

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Abstract

Leth be the Hausdorff dimension of the Julia setJ(R) of a Misiurewicz’s rational mapR :\(R:\bar {\mathbb{C}} \to \bar {\mathbb{C}}\) (subexpanding case). We prove that theh-dimensional Hausdorff measure H h onJ(R) is finite, positive and the onlyh-conformal measure forR :\(R:\bar {\mathbb{C}} \to \bar {\mathbb{C}}\) up to a multiplicative constant. Moreover, we show that there exists a uniqueR-invariant measure onJ(R) equivalent to H h .

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Authors and Affiliations

  1. Institute of Stochastics, University of Göttingen, Lotzestrasse 13, 3400, Göttingen, Germany

    M. Denker

  2. Institute of Mathematics, University of Toruń, ul. Chopina 12/18, 87-100, Toruń, Poland

    M. Urbański

Authors
  1. M. Denker
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  2. M. Urbański
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Denker, M., Urbański, M. Hausdorff measures on Julia sets of subexpanding rational maps. Israel J. Math. 76, 193–214 (1991). https://doi.org/10.1007/BF02782852

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